#276735

order	512
nilpotency class	3
isoclinism	isoclinic class of #268763
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

595 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 183931);