#6676646

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1985 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)(7,10)(8,12)(9,11);(5,7,10,6)(8,12,9,11)(13,14,17,21)(15,19,22,27)(16,18,24,20)(23,25,28,26);(2,4)(14,18)(15,22)(16,24)(19,26)(20,21)(25,27);(5,8,10,9)(6,11,7,12)(13,15,17,22)(14,19,21,27)(16,23,24,28)(18,25,20,26);(5,9,10,8)(6,11,7,12)(14,18)(16,24)(19,25)(20,21)(23,28)(26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308113, #72308867
maximal normal subgroup of	#72308113, #72308867

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6583842);