#6514430

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2074 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257074, #72257125, #72257176, …
maximal normal subgroup of	#72257074, #72257125, #72257176, …
maximal quotient of	#72265663

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6421626);