#2789028

order	512
nilpotency class	2
isoclinism	isoclinic class of #2760365
automorphism	Aut(G) order 2097152
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1684 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=gh,c^2=fh,d^2=fgh,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,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, 2696224);