#156796

order	512
nilpotency class	2
isoclinism	isoclinic class of #156684
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

2496 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=ef,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 63992);