#950052

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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

2086 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,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, 857248);