#379432

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

956 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=ef,d^2=fhi,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286628);