#386469

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

1024 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)(15,25)(16,30)(17,34)(18,24)(19,23)(20,28)(21,33)(22,31)(26,41)(27,38)(29,40)(32,42)(35,43)(36,39)(37,44);(13,15,19,29)(14,20,23,39)(16,27,33,44)(17,35,32,26)(18,36,31,28)(21,41,30,43)(22,40,24,25)(34,37,42,38);(5,7,8,12)(6,9,10,11)(13,16,31,17,19,33,18,32)(14,21,22,42,23,30,24,34)(15,26,36,44,29,35,28,27)(20,37,25,43,39,38,40,41);(6,10)(9,11)(13,17,31,33,19,32,18,16)(14,21,24,34,23,30,22,42)(15,27,36,26,29,44,28,35)(20,38,40,43,39,37,25,41)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fg,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,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, 293665);