#719698

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2540	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5636 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)(13,14,19,24)(15,21,16,20)(17,18,23,22)(25,26,27,28);(10,12)(13,15,19,16)(14,20,24,21)(17,25,23,27)(18,28,22,26);(5,6,8,7)(9,10,11,12)(13,16,19,15)(14,21,24,20)(17,25,23,27)(18,28,22,26);(1,2,3,4)(5,7,8,6)(13,17,19,23)(14,22,24,18)(15,25,16,27)(20,26,21,28);(13,18,19,22)(14,23,24,17)(15,26,16,28)(20,25,21,27)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gh,d^2=i,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, 626894);