#183022

order	512
nilpotency class	3
isoclinism	isoclinic class of #17547
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90218);