#387432

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1459 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=dei,f^c=fi,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, 294628);