#6996566

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

2419 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6903762);