#6808170

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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

2075 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)(6,9)(7,11)(15,23)(20,28)(24,27)(25,26);(5,6)(7,10)(8,9)(11,12)(13,14)(15,24)(16,21)(17,19)(18,22)(20,26)(23,27)(25,28);(2,4)(13,15,19,26,18,23,16,25)(14,20,21,24,22,28,17,27);(5,7,8,11)(6,10,9,12)(13,16,18,19)(14,21,22,17)(15,25,23,26)(20,24,28,27);(6,9)(10,12)(13,17,16,22,18,21,19,14)(15,24,25,20,23,27,26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257074, #72257078, #72257149
maximal normal subgroup of	#72257074, #72257078, #72257149
maximal quotient of	#72265647

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6715366);