#615591

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

2746 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 522787);