#4455125

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
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

1836 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)(16,23)(20,25)(21,27)(26,28)(29,30,34,31)(32,35,40,37)(33,38,41,36)(39,43,44,42);(2,4)(5,6,8,9)(7,10,11,12)(14,18)(19,24)(20,25)(26,28)(30,31)(35,37)(36,38)(42,43);(2,4)(5,7)(6,10)(8,11)(9,12)(13,14,17,18)(15,19,22,24)(16,20,23,25)(21,26,27,28)(29,31,34,30)(32,35,40,37)(33,38,41,36)(39,42,44,43);(1,2)(3,4)(5,6)(7,12)(8,9)(10,11)(13,15,17,22)(14,19,18,24)(16,21,23,27)(20,26,25,28)(29,32)(30,35)(31,37)(33,39)(34,40)(36,42)(38,43)(41,44);(13,16)(14,20)(15,21)(17,23)(18,25)(19,26)(22,27)(24,28)(29,33,34,41)(30,36,31,38)(32,39,40,44)(35,42,37,43)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=h,d^2=fgh,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, 4362321);