#181637

order	512
nilpotency class	3
isoclinism	isoclinic class of #17603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1312 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,21,27,23,28,22,26)(19,29,31,24,33,38,32,39)(20,25,35,42,34,40,36,41)(30,43,45,37,44,48,46,47);(1,3)(2,6,8,9)(4,10)(5,11)(7,13,14,15)(12,16)(17,19,22,32,23,33,21,31)(18,24,27,38,28,39,26,29)(20,30,36,46,34,44,35,45)(25,37,42,48,40,47,41,43);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,25)(19,30)(21,35)(22,36)(23,34)(24,37)(26,41)(27,42)(28,40)(29,43)(31,45)(32,46)(33,44)(38,48)(39,47);(20,34)(25,40)(30,44)(35,36)(37,47)(41,42)(43,48)(45,46)

Polycyclic presentation:

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

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, 88833);