#4944292

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

1824 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgh,c^2=fgh,d^2=fh,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, 4851488);