#192972

order	512
nilpotency class	3
isoclinism	isoclinic class of #18682
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

804 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)(4,13)(5,9)(6,10)(7,11)(8,14)(15,16)(17,18,23,19,21,25,22,27)(20,28,29,24,30,26,31,32);(1,3)(2,8)(4,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,19,23,25,21,27,22,18)(20,26,29,32,30,28,31,24);(1,4,7,8)(2,3,11,15)(5,12,6,16)(9,14,10,13)(17,20)(18,24)(19,26)(21,30)(22,31)(23,29)(25,32)(27,28);(1,5,7,6)(2,9,11,10)(3,13,15,14)(4,16,8,12)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100168);