#2240997

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1763 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=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, 2148193);