#386969

order	512
nilpotency class	3
isoclinism	isoclinic class of #30490
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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

976 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 234 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) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,3,4)(6,9)(7,11)(13,14)(15,26)(16,21)(17,33)(18,24)(19,25)(20,29)(22,32)(23,36)(27,42)(28,44)(30,38)(31,37)(34,41)(35,39)(40,43);(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,31)(17,27)(18,35)(19,30)(21,40)(22,42)(23,38)(24,26)(25,39)(28,32)(29,36)(33,37)(34,43)(41,44);(5,7,8,11)(6,10,9,12)(13,16,18,17,19,32,36,34)(14,21,24,33,25,22,23,41)(15,27,29,31,30,43,35,28)(20,37,38,40,39,44,26,42);(6,9)(10,12)(13,17,18,32,19,34,36,16)(14,22,23,33,25,21,24,41)(15,28,29,27,30,31,35,43)(20,37,26,42,39,44,38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fg,d^2=ef,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,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294165);