#7501734

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2570 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308191, #72308252, #72308276
maximal normal subgroup of	#72308191, #72308252, #72308276

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408930);