#832801

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2304 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)(3,6,7,8)(9,10,14,13)(11,15,12,16)(17,18,22,21)(19,24,20,23)(25,26)(27,28)(29,31)(30,32);(2,4)(6,8)(10,13)(15,16)(25,27,29,32)(26,30,31,28);(9,11,14,12)(10,15,13,16)(17,19,22,20)(18,23,21,24)(25,26)(27,28)(29,31)(30,32);(1,3)(2,6)(4,8)(5,7)(9,12,14,11)(10,15,13,16)(17,20,22,19)(18,24,21,23)(25,26)(27,30)(28,32)(29,31);(1,4,5,2)(3,6,7,8)(9,13,14,10)(11,16,12,15)(17,21,22,18)(19,24,20,23)(25,28,29,30)(26,27,31,32)

Polycyclic presentation:

2^9|a^2=fgh,b^2=h,c^2=i,d^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 739997);