#946368

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

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

4398 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 853564);