#7021468

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2260 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6928664);