#865723

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fgh,c^2=gi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 772919);