#4456856

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
automorphism	Aut(G) order 2097152
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1993 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)(6,9)(7,11)(15,21)(16,23)(19,24)(20,25)(30,34)(31,37)(36,41)(38,43);(1,2,3,4)(5,6)(7,12)(8,9)(10,11)(14,18)(19,24)(20,25)(26,28)(31,37)(35,40)(38,43)(42,44);(1,2,3,4)(5,6)(7,12)(8,9)(10,11)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,30)(31,35)(32,36)(33,34)(37,40)(38,42)(39,41)(43,44);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,31)(30,35)(32,38)(33,37)(34,40)(36,42)(39,43)(41,44);(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,32,33,39)(30,36,34,41)(31,38,37,43)(35,42,40,44)

Polycyclic presentation:

2^9|b^2=ghi,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, 4364052);