#7149192

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

2175 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:

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

Polycyclic presentation:

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

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, 7056388);