#423652

order	512
nilpotency class	3
isoclinism	isoclinic class of #417859
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1624 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330848);