#4462494

order	512
nilpotency class	2
isoclinism	isoclinic class of #4462490
automorphism	Aut(G) order 8388608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4264 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)(6,9)(7,11)(15,19)(18,20)(22,23)(24,28);(6,9)(10,12)(14,17)(18,20)(21,22,25,23)(24,26,28,27);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23,25,22)(24,26,28,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(21,24)(22,26)(23,27)(25,28);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)

Polycyclic presentation:

2^9|b^2=f,c^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328049, #72328055, #72328067
maximal normal subgroup of	#72328049, #72328055, #72328067
maximal quotient of	#72331418

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4369690);