#4462489

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1840 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)(10,12)(13,14,18,17)(15,21,23,19)(16,20,24,25)(22,27,28,26)(29,30,34,32)(31,37,39,35)(33,40,41,36)(38,42,44,43);(2,4)(16,24)(20,25)(22,28)(26,27)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(2,4)(13,15,18,23)(14,19,17,21)(16,22,24,28)(20,26,25,27)(29,31)(30,35)(32,37)(33,38)(34,39)(36,42)(40,43)(41,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,24)(14,20,17,25)(15,22,23,28)(19,26,21,27)(29,32,34,30)(31,35,39,37)(33,40,41,36)(38,42,44,43);(5,7,8,11)(6,10,9,12)(13,17,18,14)(15,21,23,19)(16,25,24,20)(22,27,28,26)(29,33,34,41)(30,36,32,40)(31,38,39,44)(35,42,37,43)

Polycyclic presentation:

2^9|a^2=fgh,b^2=h,c^2=fg,d^2=fgh,e^2=fghi,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, 4369685);