#688037

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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×#2407	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5212 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)(5,6,7,8)(10,12)(14,19)(16,23)(21,25)(24,28);(10,12)(13,14,18,19)(15,16,20,23)(17,25,26,21)(22,28,27,24);(5,6)(7,8)(9,10,11,12)(13,15)(14,16)(17,22)(18,20)(19,23)(21,24)(25,28)(26,27);(1,2)(3,4)(5,6)(7,8)(13,16)(14,20)(15,19)(17,24)(18,23)(21,27)(22,25)(26,28);(13,17,18,26)(14,21,19,25)(15,22,20,27)(16,24,23,28)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=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, 595233);