#4936018

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 2097152
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

1682 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)(5,6)(7,11)(8,12)(9,10)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26)(29,30)(31,32)(33,37)(34,35)(36,38)(39,40)(41,42)(43,44);(5,7,10,8)(6,11,9,12)(15,21)(19,22)(23,27)(26,28)(30,35)(31,38)(33,41)(40,43);(5,8,10,7)(6,11,9,12)(13,15,18,21)(14,19,16,22)(17,23,25,27)(20,26,24,28)(29,31)(30,36)(32,35)(33,39)(34,38)(37,43)(40,42)(41,44);(1,2)(3,4)(5,9,10,6)(7,12,8,11)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26)(29,32,34,36)(30,31,35,38)(33,40,41,43)(37,39,42,44);(13,17)(14,20)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28)(29,33,34,41)(30,37,35,42)(31,39,38,44)(32,40,36,43)

Polycyclic presentation:

2^9|a^2=fh,b^2=fg,c^2=gh,d^2=fgh,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, 4843214);