#384363

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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

1026 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)(7,11)(8,10)(9,12)(13,14)(15,24)(16,30)(17,35)(18,22)(19,23)(20,27)(21,32)(25,29)(26,41)(28,37)(31,38)(33,39)(34,44)(36,43)(40,42);(13,15,19,29)(14,20,23,37)(16,31,34,42)(17,28,18,27)(21,38,39,40)(22,25,35,24)(26,33,43,32)(30,36,44,41);(5,7,8,12)(6,9,10,11)(13,14)(15,25)(16,21)(17,22)(18,35)(19,23)(20,27)(24,29)(26,40)(28,37)(30,32)(31,36)(33,44)(34,39)(38,43)(41,42);(6,10)(9,11)(13,16,18,33,19,34,17,32)(14,21,35,44,23,39,22,30)(15,26,28,42,29,43,27,31)(20,36,25,40,37,41,24,38)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,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,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 291559);