#6576613

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
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 ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2827 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281943, #72282179, #72282498
maximal normal subgroup of	#72281943, #72282179, #72282498

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483809);