#386025

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1586 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1586 conjugacy classes of subgroups — too many to draw. Only its 274 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,4)(5,6,7,8)(9,10)(11,21)(12,18)(13,29)(14,30)(15,19)(16,20)(17,24)(22,33)(23,38)(25,32)(26,35)(27,39)(28,34)(31,36)(37,40);(9,11,16,26)(10,17,20,32)(12,23,28,36)(13,22,27,37)(14,25,15,24)(18,33,34,40)(19,35,30,21)(29,31,39,38);(2,4)(9,12,14,27,16,28,15,13)(10,18,30,39,20,34,19,29)(11,22,24,36,26,37,25,23)(17,31,35,40,32,38,21,33);(2,4)(9,13,15,28,16,27,14,12)(10,18,30,39,20,34,19,29)(11,23,25,37,26,36,24,22)(17,31,35,40,32,38,21,33)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi

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, 293221);