#6459999

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2567 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307390, #72307404, #72307552
maximal normal subgroup of	#72307390, #72307404, #72307552

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6367195);