#369297

order	512
nilpotency class	3
isoclinism	isoclinic class of #196350
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=efg,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi

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, 276493);