#6571623

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533105
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4074 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)(7,9)(8,10)(11,12);(6,9)(7,12)(10,11)(13,14,17,20)(15,21,23,18)(16,19,24,26)(22,27,28,25);(2,4)(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,25,26,27);(6,10)(9,11)(13,16)(14,19)(15,22)(17,24)(18,25)(20,26)(21,27)(23,28);(16,24)(19,26)(22,28)(25,27)

Polycyclic presentation:

2^9|b^2=h,c^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,f^b=fi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279976
maximal normal subgroup of	#72279976
maximal quotient of	#72280665, #72280749

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6478819);