#404876

order	512
nilpotency class	3
isoclinism	isoclinic class of #393909
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(21,22,25,30)(23,31,32,35)(24,28,26,29)(27,34,36,33);(2,4)(5,7,9,8)(6,10,11,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,27)(24,34)(25,32)(26,33)(28,31)(29,35)(30,36);(2,4)(5,8,9,7)(6,10,11,12)(14,17)(18,20)(22,28)(23,32)(24,26)(27,35)(29,30)(31,36);(13,15,16,19)(14,18,17,20)(22,29)(24,26)(27,35)(28,30)(31,36)(33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312072);