#362662

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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

1324 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efg,c^2=i,d^2=efi,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,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=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, 269858);