#207054

order	512
nilpotency class	3
isoclinism	isoclinic class of #162406
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

834 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)(17,18,21,23,22,26,20,25)(19,24,27,30,28,32,29,31);(2,5)(6,7)(9,10,11,15,12,14,16,13)(18,23,26,25)(24,30,32,31);(1,3,4,8)(2,6,5,7)(17,19)(18,24)(20,29)(21,27)(22,28)(23,30)(25,31)(26,32);(17,20,22,21)(18,25,26,23)(19,27,28,29)(24,30,32,31)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 114250);