#417714

order	512
nilpotency class	3
isoclinism	isoclinic class of #417700
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

2473 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)(7,16)(8,13)(9,24)(10,15)(11,19)(12,22)(14,26)(17,27)(18,34)(20,33)(21,31)(23,30)(25,29)(28,36)(32,35);(5,7)(6,11)(8,17)(9,25)(10,20)(12,28)(13,32)(14,33)(15,29)(16,24)(18,21)(19,26)(22,35)(23,36)(27,31)(30,34);(5,8)(6,12)(7,17)(9,21)(10,23)(11,27)(13,24)(14,30)(15,31)(16,34)(18,25)(19,35)(20,36)(22,26)(28,29)(32,33);(2,4)(5,8)(6,13)(7,18)(9,22)(10,23)(11,28)(12,24)(14,31)(15,30)(16,34)(17,25)(19,36)(20,35)(21,26)(27,29)(32,33)

Polycyclic presentation:

2^9|e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,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, 324910);