#1097894

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2466 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=gh,d^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1005090);