#6342406

order	512
nilpotency class	2
isoclinism	isoclinic class of #6342371
automorphism	Aut(G) order 16777216
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2832 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219274, #72219277
maximal normal subgroup of	#72219274, #72219277

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6249602);