#6499784

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499780
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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 ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4395 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)(15,22)(16,20)(17,25)(18,21)(19,24)(23,28)(26,27);(5,6)(7,11)(8,10)(9,12)(13,15)(14,19)(16,24)(17,26)(18,23)(20,28)(21,27)(22,25);(2,4)(5,7)(6,9)(8,12)(10,11)(15,23)(19,27)(22,28)(24,26);(7,12)(9,11)(13,16,18,17)(14,20,21,25)(15,24,23,26)(19,28,27,22);(6,10)(9,11)(15,23)(19,27)(22,28)(24,26)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280747, #72280757, #72280990, …
maximal normal subgroup of	#72280747, #72280757, #72280990, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6406980);