#331701

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#24116	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

793 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)(17,18,21,26)(19,25,22,23)(20,28,29,31)(24,30,32,27);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,15)(14,16)(17,19,21,22)(18,23,26,25)(20,27,29,30)(24,28,32,31);(10,13)(11,16)(14,15)(17,20,21,29)(18,24,26,32)(19,27,22,30)(23,28,25,31);(2,6)(3,8)(18,25)(19,22)(23,26)(24,31)(27,30)(28,32)

Polycyclic presentation:

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

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, 238897);