#183271

order	512
nilpotency class	3
isoclinism	isoclinic class of #17603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

936 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,d^a=dh,e^a=eh,g^a=ghi,e^b=ei,d^c=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90467);