#159186

order	512
nilpotency class	2
isoclinism	isoclinic class of #158815
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

3072 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

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, 66382);