#7145751

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

2244 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)(13,14,18,22,19,23,17,20)(15,24,26,28,25,16,21,27);(6,9)(7,11)(14,20)(15,25)(16,27)(17,18)(22,23)(24,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,21)(18,26)(19,25)(20,28)(22,27)(23,24);(6,9)(10,12)(13,16,18,27,19,24,17,28)(14,21,22,15,23,26,20,25);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,22,23,20)(15,26,25,21)(16,28,24,27)

Polycyclic presentation:

2^9|a^2=f,d^2=f,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=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, 7052947);