#4327956

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 4194304
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	C₂×#2404	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3632 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)(5,6)(7,11)(8,9)(10,12)(13,14,17,19)(15,20,18,16)(21,22,24,26)(23,25,27,28);(6,9)(10,11)(13,15)(14,16)(17,18)(19,20)(23,27)(25,28);(1,2)(3,4)(7,12)(10,11)(15,18)(16,20)(21,23,24,27)(22,25,26,28);(5,7,8,12)(6,10,9,11)(13,16,17,20)(14,18,19,15)(21,22)(23,25)(24,26)(27,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,18)(17,19)(21,22)(23,25)(24,26)(27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4235152);