#408252

order	512
nilpotency class	3
isoclinism	isoclinic class of #393988
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1071 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)(3,5)(4,6)(7,8)(9,10)(11,16)(12,24,26,37)(13,28)(14,19)(15,20)(17,27,36,30)(18,29)(21,31)(22,35,38,40)(23,33)(25,32,39,34);(1,3,4,7)(2,5,6,8)(9,11,21,29,15,23,14,13)(10,16,31,18,20,33,19,28)(12,22,27,39,26,38,30,25)(17,34,24,40,36,32,37,35);(9,12)(10,17)(11,22)(13,25)(14,30)(15,26)(16,32)(18,35)(19,37)(20,36)(21,27)(23,38)(24,31)(28,40)(29,39)(33,34);(2,6)(5,8)(9,13,14,23,15,29,21,11)(10,18,19,16,20,28,31,33)(12,25,30,38,26,39,27,22)(17,35,37,32,36,40,24,34)

Polycyclic presentation:

2^9|a^2=e,b^2=efh,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^c=ei,f^c=fi

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, 315448);