#207038

order	512
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,d^2=h,e^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,c^b=ch,e^b=ei,g^b=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 114234);