#7146145

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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

2220 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)(3,4)(7,11)(10,12)(13,14,19,20,18,24,17,23)(15,21,27,16,22,28,25,26);(6,9)(7,11)(14,20)(16,21)(17,19)(23,24)(25,27)(26,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,20)(17,25)(18,22)(19,27)(23,26)(24,28);(6,9)(10,12)(13,16,19,28,18,26,17,21)(14,22,20,25,24,15,23,27);(5,7,8,11)(6,10,9,12)(15,22)(16,26)(21,28)(25,27)

Polycyclic presentation:

2^9|a^2=fi,d^2=fi,e^2=h,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=dhi,e^c=ei,e^d=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, 7053341);