#1082698

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

1883 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,d^2=fi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef

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, 989894);