#152079

order	512
nilpotency class	5
isoclinism	isoclinic class of #152078
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12695
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#124	C₂³
prime radical	#535	D₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #117 ◁ #3510 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

586 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dh,f^b=fghi,g^b=gi,e^c=ei,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, 59275);