#213000

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

676 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=fh,d^2=gh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ci,d^c=dh

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, 120196);