#386282

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

770 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gi,d^2=e,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, 293478);