#3069035

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1976 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)(27,28)(29,37)(30,31)(32,35)(33,38)(34,39)(36,40);(1,3,4,8)(2,5,7,6)(9,11)(10,15)(12,16)(13,14)(17,19)(18,22)(20,23)(21,24)(26,31)(28,32)(29,38)(34,40);(9,10,14,12)(11,16,13,15)(17,18,21,20)(19,22,24,23)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,37)(38,40);(1,2)(3,5)(4,7)(6,8)(9,12,14,10)(11,16,13,15)(17,19)(18,22)(20,23)(21,24)(25,28,30,32)(26,27,31,35)(29,39,38,36)(33,40,37,34);(1,3)(2,6)(4,8)(5,7)(9,13,14,11)(10,15,12,16)(17,20,21,18)(19,22,24,23)(25,29,30,38)(26,33,31,37)(27,34,35,40)(28,36,32,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=fghi,c^2=hi,d^2=gh,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, 2976231);