#3069032

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1938 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fhi,c^2=hi,d^2=h,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef

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, 2976228);