#182965

order	512
nilpotency class	3
isoclinism	isoclinic class of #17597
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 ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

921 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)(3,12)(4,10)(5,9)(6,11)(7,13)(8,14)(15,16)(17,18,22,25,23,19,21,27)(20,26,29,32,30,24,28,31);(1,3)(2,7,11,16)(4,15)(5,13)(6,14)(8,10,12,9)(17,19,23,18)(20,26,30,24)(21,25,22,27)(28,31,29,32);(1,3)(2,8)(4,15)(5,13)(6,14)(7,10)(9,16)(11,12)(17,20,22,29,23,30,21,28)(18,24,25,31,19,26,27,32);(1,4,6,5)(2,9,11,10)(3,13,14,15)(7,12,16,8)(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,d^c=dhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90161);