#4342843

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1674 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=fh,d^2=fhi,e^2=fg,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, 4250039);