#417679

order	512
nilpotency class	3
isoclinism	isoclinic class of #417534
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2420 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)(6,12)(9,10)(13,27)(14,26)(15,16)(17,29)(19,25)(21,30)(22,24)(28,35)(31,36)(32,33);(5,6)(7,17)(8,21)(9,16)(10,12)(11,15)(13,19)(14,22)(18,28)(20,27)(23,26)(24,30)(25,29)(31,32)(33,35)(34,36);(5,7)(6,13)(8,18)(9,19)(10,25)(11,20)(12,17)(14,28)(15,29)(16,27)(21,35)(22,32)(23,34)(24,33)(26,31)(30,36);(2,4)(5,8,11,23)(6,14,15,30)(7,18,20,34)(9,24,10,22)(12,26,16,21)(13,28,29,36)(17,31,27,35)(19,33,25,32)

Polycyclic presentation:

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

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, 324875);