#361397

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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

972 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)(13,14)(15,20)(16,19)(17,18)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(5,7,9,8)(6,10,11,12)(14,18)(19,20)(21,23)(22,26)(24,31)(25,30)(27,29)(28,34)(32,33)(35,36);(5,8,9,7)(6,10,11,12)(13,15,17,16)(14,19,18,20)(23,30)(25,33)(26,34)(28,36)(29,35)(31,32);(6,11)(10,12)(13,16,17,15)(14,20,18,19)(22,27)(26,29)(28,36)(34,35)

Polycyclic presentation:

2^9|a^2=h,b^2=efi,c^2=ei,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,g^c=gi

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, 268593);