#2044084

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
automorphism	?
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1604 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=fh,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=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, 1951280);