#386907

order	512
nilpotency class	3
isoclinism	isoclinic class of #30490
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	D₄×C₄⋊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

880 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)(3,5)(4,7)(6,8)(9,10,14,16,13,18,28,19)(11,20,21,31,23,15,24,32)(12,25,29,35,27,34,40,17)(22,37,36,30,26,39,38,33);(1,3)(2,4)(5,8)(6,7)(9,11)(10,15)(12,26)(13,23)(14,21)(16,32)(17,33)(18,20)(19,31)(22,27)(24,28)(25,37)(29,38)(30,35)(34,39)(36,40);(1,4,6,5)(2,7,8,3)(10,16,18,19)(11,21,23,24)(12,27)(15,20)(17,34,35,25)(22,38,26,36)(29,40)(31,32);(1,5,6,4)(2,7,8,3)(9,12,28,40,13,27,14,29)(10,17,16,25,18,35,19,34)(11,22,24,38,23,26,21,36)(15,30,32,39,20,33,31,37)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294103);