#6808177

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

2093 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)(15,22)(20,28)(23,25)(24,26);(5,6)(7,10)(8,9)(11,12)(13,14,18,21)(15,23,22,25)(16,17,19,27)(20,24,28,26);(2,4)(13,15,19,26,18,22,16,24)(14,20,17,23,21,28,27,25);(5,7,8,11)(6,10,9,12)(13,16,18,19)(14,17,21,27)(15,24,22,26)(20,23,28,25);(6,9)(10,12)(13,17,19,21,18,27,16,14)(15,25,26,20,22,23,24,28)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=fi,e^2=h,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	#72256862, #72256890, #72256932, …
maximal normal subgroup of	#72256862, #72256890, #72256932, …
maximal quotient of	#72264220, #72265102

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6715373);