#6583777

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1953 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282046, #72282092
maximal normal subgroup of	#72282046, #72282092

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490973);