#6403432

order	512
nilpotency class	3
isoclinism	isoclinic class of #62283
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1769 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^b=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6310628);