#387949

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

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

1540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=eg,d^2=eg,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=dei,f^c=fi,f^d=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, 295145);