#210636

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
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	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

686 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)(7,10)(8,11)(9,13)(12,16)(14,18)(15,17)(19,20)(21,22,25,24,26,28,33,32)(23,27,31,30,29,34,36,35);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,17)(12,19)(13,18)(16,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,35)(33,36);(6,11)(8,16)(10,17)(12,13)(15,20)(18,19)(23,29)(27,34)(30,35)(31,36);(7,14)(10,18)(15,20)(17,19)(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=e,d^2=g,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ci,d^b=dh,f^c=fh

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, 117832);