#6570672

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2444 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=gi,d^2=hi,e^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284128, #72308966, #72309020
maximal normal subgroup of	#72284128, #72308966, #72309020

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6477868);