#182937

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

966 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)(3,11)(4,10)(5,9)(6,8)(7,12)(13,16)(14,15)(17,18,22,25,23,19,21,27)(20,26,29,32,30,24,28,31);(1,3)(2,7,8,15)(4,14)(5,12)(6,13)(9,16,10,11)(17,19,23,18)(20,26,30,24)(21,25,22,27)(28,31,29,32);(2,8)(7,15)(9,10)(11,16)(17,20,22,29,23,30,21,28)(18,24,25,31,19,26,27,32);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)(17,21,23,22)(18,25,19,27)(20,28,30,29)(24,31,26,32)

Polycyclic presentation:

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

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, 90133);