#6677413

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667676
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 ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2137 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305239, #72305243, #72305279, …
maximal normal subgroup of	#72305239, #72305243, #72305279, …
maximal quotient of	#72305527, #72305591

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6584609);