#4942014

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2494 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=fh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

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, 4849210);