#386465

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
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

1034 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)(16,31)(17,34)(18,35)(19,24)(20,23)(21,28)(22,33)(26,41)(27,36)(29,40)(30,37)(32,39)(38,43)(42,44);(13,15,19,29)(14,21,24,37)(16,27,33,42)(17,26,32,43)(18,30,20,28)(22,38,31,41)(23,40,35,25)(34,44,39,36);(5,7,8,12)(6,9,10,11)(13,16,18,32,19,33,20,17)(14,22,35,34,24,31,23,39)(15,26,28,42,29,43,30,27)(21,36,40,41,37,44,25,38);(6,10)(9,11)(13,17,20,33,19,32,18,16)(14,22,35,34,24,31,23,39)(15,27,30,43,29,42,28,26)(21,36,40,41,37,44,25,38)

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

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, 293661);