#616972

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2613	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

6125 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 524168);