#367852

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2358
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

972 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)(4,5)(6,8)(9,10,14,21,16,22,15,20)(11,23,26,19,13,33,25,17)(12,27,31,37,32,38,30,18)(24,34,28,35,29,36,39,40);(2,5)(6,7)(9,11)(10,17)(12,28,32,39)(13,16)(14,26)(15,25)(18,35,37,40)(19,22)(20,33)(21,23)(24,31,29,30)(27,36,38,34);(9,12)(10,18)(11,24)(13,29)(14,31)(15,30)(16,32)(17,34)(19,36)(20,38)(21,27)(22,37)(23,35)(25,39)(26,28)(33,40);(1,3,4,8)(2,6,5,7)(9,13,16,11)(10,19,22,17)(12,29,32,24)(14,25,15,26)(18,36,37,34)(20,23,21,33)(27,40,38,35)(28,31,39,30)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=ef,e^2=i,f^2=i,g^2=i,h^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, 275048);