#210712

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	#2011	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

758 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=g,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=dhi,d^c=dhi,f^c=fh

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, 117908);