#152132

order	512
nilpotency class	5
isoclinism	isoclinic class of #152132
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12698
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

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

normal subgroups only

This group has 538 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)(3,13,17,39)(4,18)(5,22,24,55)(6,26,28,11)(7,12)(8,15,31,44)(9,20)(10,25)(14,37,42,61)(16,38,27,40)(19,48,50,54)(21,35,52,47)(23,32,45,30)(29,41,57,63)(33,49,58,64)(34,60)(36,56)(43,59,53,62)(46,51);(1,3)(2,8,12,30)(4,14)(5,23)(6,27)(7,16)(9,33,18,29)(10,13,36,38)(11,31,26,32)(15,24)(17,28)(19,49)(20,43)(21,53)(22,39,55,40)(25,45)(34,59,46,37)(35,58,47,57)(41,50)(42,52)(44,56)(48,61,54,62)(51,64)(60,63);(1,4)(2,9,12,18)(3,14)(5,19)(6,21)(7,20)(8,29,30,33)(10,34,36,46)(11,35,26,47)(13,37,38,59)(15,41)(16,43)(17,42)(22,54,55,48)(23,49)(24,50)(25,51)(27,53)(28,52)(31,57,32,58)(39,61,40,62)(44,63)(45,64)(56,60)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ch,d^b=dhi,e^b=ei,f^b=fghi,g^b=gi,e^c=ei,f^d=fhi,g^f=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59328);