#4308619

order	512
nilpotency class	2
isoclinism	isoclinic class of #4307937
automorphism	Aut(G) order 134217728
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

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

2098 conjugacy classes of subgroups; only the 738 normal subgroups are shown.

normal subgroups only

This group has 2098 conjugacy classes of subgroups — too many to draw. Only its 738 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(7,11)(9,12)(13,14,18,15)(16,20,17,19)(24,31)(28,33)(29,35)(34,36);(1,2,3,4)(5,6,8,10)(7,9,11,12)(13,15,18,14)(16,20,17,19)(22,26)(27,32)(28,33)(34,36);(13,16,18,17)(14,19,15,20)(21,22,25,26)(23,27,30,32)(24,28,31,33)(29,34,35,36);(5,7,8,11)(6,9,10,12)(13,17,18,16)(14,20,15,19)(21,23,25,30)(22,27,26,32)(24,29,31,35)(28,34,33,36);(21,24,25,31)(22,28,26,33)(23,29,30,35)(27,34,32,36)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=gh,d^2=hi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4215815);