#182964

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	#3444	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

915 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)(3,10)(4,13)(5,8)(6,12)(7,11)(9,14)(15,16)(17,18,22,25,23,19,21,27)(20,26,29,32,30,24,28,31);(1,3)(2,6,8,15)(4,12)(5,11)(7,9,10,13)(14,16)(17,19,23,18)(20,26,30,24)(21,25,22,27)(28,31,29,32);(1,3)(2,7)(4,12)(5,11)(6,9)(8,10)(13,15)(14,16)(17,20,22,29,23,30,21,28)(18,24,25,31,19,26,27,32);(2,8)(3,11)(9,13)(12,16)(18,19)(24,26)(25,27)(31,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,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, 90160);