#372624

order	512
nilpotency class	3
isoclinism	isoclinic class of #222240
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1117 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)(3,7)(4,5)(6,8)(9,10)(11,21)(12,25,29,35)(13,18)(14,19)(15,20)(16,23)(17,28,34,27)(22,37,26,31)(24,32)(30,33)(36,38,40,39);(2,5)(6,7)(9,11)(10,16)(12,26,29,22)(13,30)(14,23)(15,24)(17,31,34,37)(18,21)(19,32)(20,33)(25,36,35,40)(27,38,28,39);(9,12)(10,17)(11,22)(13,28)(14,27)(15,29)(16,31)(18,25)(19,35)(20,34)(21,36)(23,39)(24,26)(30,38)(32,40)(33,37);(1,3,4,8)(2,6,5,7)(10,18)(13,14)(16,21)(17,25)(19,20)(23,30)(27,28)(31,36)(32,33)(34,35)(37,40)(38,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=efg,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,f^c=fi,h^c=hi,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, 279820);