#6596085

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573549
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#36157	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3354 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)(13,14,18,26,17,20,16,21)(15,22,25,19,24,28,23,27);(5,6,8,9)(7,10,11,12)(13,15,17,24)(14,19,20,27)(16,25,18,23)(21,28,26,22);(2,4)(6,9)(7,11);(5,7)(6,10)(8,11)(9,12)(14,20)(19,27)(21,26)(22,28);(6,9)(7,11)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307480, #72307620, #72308056
maximal normal subgroup of	#72307480, #72307620, #72308056

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6503281);