#290896

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
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	#29814	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

814 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)(3,7)(4,6)(5,8)(9,10,13,15)(11,16,14,12)(17,18,23,26)(19,24,30,27)(20,21,25,22)(28,32,29,31);(1,3,4,8)(2,5,6,7)(11,14)(12,16)(17,19)(18,24)(20,31)(21,28)(22,29)(23,30)(25,32)(26,27);(9,11,12,10,13,14,16,15)(17,20,22,18,23,25,21,26)(19,27,29,31,30,24,28,32);(2,6)(5,7)(17,21,23,22)(18,25,26,20)(19,28,30,29)(24,32,27,31)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198092);