#336255

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#28140	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

710 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)(3,4)(5,7)(6,8)(9,10)(11,13)(12,15)(14,16)(17,18,20,23)(19,25,27,22)(21,24,29,32)(26,31,28,30);(1,3)(2,6)(4,7)(5,8)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,29)(31,32);(1,4,5,6)(2,3,7,8)(10,13)(11,16)(14,15)(19,26)(21,29)(22,30)(24,32)(25,31)(27,28);(2,7)(3,8)(10,13)(11,16)(14,15)(19,27)(22,25)(26,28)(30,31)

Polycyclic presentation:

2^9|a^2=i,c^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 243451);