#374748

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

1062 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fh,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=di,g^c=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, 281944);