#372629

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	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1096 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)(11,22)(12,26,30,17)(13,31)(14,21)(15,20)(16,24)(18,25)(19,33)(23,39,28,35)(27,34,40,36)(29,37,32,38);(2,5)(6,7)(9,11)(10,16)(12,27,30,40)(13,15)(14,25)(17,35,26,39)(18,20)(19,31)(21,22)(23,29,28,32)(24,33)(34,38,36,37);(9,12)(10,17)(11,23)(13,28)(14,32)(15,30)(16,34)(18,36)(19,38)(20,26)(21,37)(22,39)(24,27)(25,40)(29,33)(31,35);(1,3,4,8)(2,6,5,7)(9,13,15,11)(10,18,20,16)(12,28,30,23)(14,24,33,25)(17,36,26,34)(19,22,21,31)(27,29,40,32)(35,38,39,37)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=efg,e^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,g^c=gi,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, 279825);