#415751

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1040 conjugacy classes of subgroups; only the 290 normal subgroups are shown.

normal subgroups only

This group has 1040 conjugacy classes of subgroups — too many to draw. Only its 290 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 — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=f,d^2=ghi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322947);