#6676652

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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

2115 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)(14,19)(15,23)(16,25)(20,22);(5,7,10,6)(8,12,9,11)(13,14,17,19)(15,21,23,27)(16,26,25,28)(18,20,24,22);(2,4)(14,20)(15,23)(18,24)(19,22)(21,28)(26,27);(5,8,10,9)(6,11,7,12)(13,15,17,23)(14,21,19,27)(16,24,25,18)(20,26,22,28);(5,9,10,8)(6,11,7,12)(13,16,17,25)(14,21,19,27)(15,24,23,18)(20,28,22,26)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308261, #72308851
maximal normal subgroup of	#72308261, #72308851

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6583848);