#4456854

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1922 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)(8,12)(10,11)(15,21)(16,23)(19,24)(20,25)(29,30,34,37)(31,32,38,35)(33,41,42,36)(39,43,44,40);(1,2,3,4)(5,6,9,7)(8,10,12,11)(14,18)(19,24)(20,25)(26,28)(31,38)(32,35)(39,44)(40,43);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31)(30,35)(32,37)(33,39)(34,38)(36,43)(40,41)(42,44);(1,2)(3,4)(5,8)(6,10)(7,11)(9,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32,34,35)(30,31,37,38)(33,40,42,43)(36,39,41,44);(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,42)(30,36,37,41)(31,39,38,44)(32,40,35,43)

Polycyclic presentation:

2^9|a^2=fg,b^2=fhi,c^2=gi,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,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, 4364050);