#6598386

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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
prime radical	#38080	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2091 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:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307555, #72307677, #72308056
maximal normal subgroup of	#72307555, #72307677, #72308056

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6505582);