#2633412

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
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

1825 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=i,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2540608);