#4936495

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6,8,9)(7,10,11,12)(13,14,18,19)(15,22,20,16)(17,21,25,26)(23,28,27,24)(30,32)(33,41)(35,39)(37,43);(14,19)(15,20)(17,25)(24,28)(29,30,34,32)(31,35,38,39)(33,40,41,36)(37,44,43,42);(6,9)(10,12)(13,15,18,20)(14,16,19,22)(17,23,25,27)(21,24,26,28)(29,31)(30,35)(32,39)(33,37)(34,38)(36,42)(40,44)(41,43);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,19)(17,24)(18,22)(21,27)(23,26)(25,28)(29,32,34,30)(31,35,38,39)(33,40,41,36)(37,42,43,44);(5,7,8,11)(6,10,9,12)(13,17)(14,21)(15,23)(16,24)(18,25)(19,26)(20,27)(22,28)(29,33,34,41)(30,36,32,40)(31,37,38,43)(35,42,39,44)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=h,d^2=fh,e^2=fg,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, 4843691);