#6445376

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305366, #72305367, #72305376
maximal normal subgroup of	#72305366, #72305367, #72305376
maximal quotient of	#72305885, #72306063

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352572);