#417801

order	512
nilpotency class	3
isoclinism	isoclinic class of #417579
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₄

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

2696 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)(5,6,12,17)(7,18,22,34)(8,9,14,27)(10,16,11,15)(13,29,32,21)(19,25,36,35)(20,23,31,33)(24,28,30,26);(5,7,10,21,12,22,11,29)(6,13,15,18,17,32,16,34)(8,23,24,19,14,33,30,36)(9,25,26,31,27,35,28,20);(5,8,12,14)(6,9,17,27)(7,19,22,36)(10,24,11,30)(13,31,32,20)(15,26,16,28)(18,35,34,25)(21,33,29,23);(2,4)(5,9)(6,14)(7,20)(8,17)(10,28)(11,26)(12,27)(13,19)(15,24)(16,30)(18,23)(21,35)(22,31)(25,29)(32,36)(33,34)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^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,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, 324997);