#152758

order	512
nilpotency class	5
isoclinism	isoclinic class of #149927
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#124	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

561 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6)(7,17,22,39)(8,24,26,13)(9,28,18,46)(10,32,33,16)(11,15)(12,20,35,47)(14,31)(19,43,51,67)(21,45,23,44)(25,57)(27,38)(29,36,53,42)(30,62)(34,49,65,58)(37,59,40,63)(41,48)(50,64,54,68)(52,66,61,55)(56,60);(5,7,18,47,11,21,9,29)(6,12,14,39,15,36,41,44)(8,19,25,58,26,50,30,61)(10,23,31,53,33,22,48,20)(13,34,40,68,24,55,37,67)(16,42,46,17,32,35,28,45)(27,51,63,49,60,54,59,52)(38,65,62,43,56,66,57,64);(2,4)(5,8)(6,13)(7,19)(9,30)(10,27)(11,26)(12,34)(14,40)(15,24)(16,38)(17,43)(18,25)(20,52)(21,50)(22,54)(23,51)(28,57)(29,61)(31,63)(32,56)(33,60)(35,66)(36,55)(37,41)(39,68)(42,65)(44,67)(45,64)(46,62)(47,58)(48,59)(49,53)

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dghi,e^b=ei,f^b=fghi,g^b=ghi,h^b=hi,d^c=di,f^c=fi,f^d=fhi,g^f=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59954);