#4315918

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
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

1822 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fh,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,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, 4223114);