#369861

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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

1130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=g,d^2=efi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,d^c=di

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, 277057);