#256698

order	512
nilpotency class	3
isoclinism	isoclinic class of #246503
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

823 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,8)(4,6)(5,7)(9,10,14,16)(11,12,15,13)(17,18,20,24)(19,26,27,22)(21,23,29,25)(28,32,30,31);(1,3,5,4)(2,6,7,8)(9,11)(10,12)(13,16)(14,15)(17,19,20,27)(18,22,24,26)(21,30,29,28)(23,32,25,31);(1,4,5,3)(2,6,7,8)(10,15)(11,16)(12,13)(18,23)(21,29)(22,31)(24,25)(26,32)(28,30);(2,7)(6,8)(9,12,14,13)(10,15,16,11)(18,24)(22,26)(23,25)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=ehi,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 163894);