#829503

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
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	#14722	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2563 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)(11,12,16,15)(17,18,20,22)(19,21,24,23)(25,26)(27,28)(29,31)(30,32);(2,4)(6,8)(9,11,14,16)(10,15,13,12)(17,18)(19,23)(20,22)(21,24)(25,27,29,32)(26,30,31,28);(17,18)(19,21)(20,22)(23,24)(25,26)(27,28)(29,31)(30,32);(1,3,5,7)(2,6,4,8)(9,12)(10,11)(13,16)(14,15)(17,18)(19,23)(20,22)(21,24)(25,26)(27,30)(28,32)(29,31);(1,4,5,2)(3,6,7,8)(9,13,14,10)(11,15,16,12)(17,19)(18,21)(20,24)(22,23)(25,28,29,30)(26,27,31,32)

Polycyclic presentation:

2^9|a^2=fh,b^2=ghi,d^2=fhi,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, 736699);