#417857

order	512
nilpotency class	3
isoclinism	isoclinic class of #417601
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

2715 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)(8,21)(9,25)(10,15)(11,18)(12,23)(13,20)(14,26)(17,29)(19,28)(22,32)(24,31)(27,35)(30,36)(33,34);(5,7,9,18,10,20,26,19)(6,11,14,16,15,28,25,13)(8,17,22,34,24,35,23,36)(12,29,31,30,32,27,21,33);(5,8,10,24)(6,12,15,32)(7,17,20,35)(9,22,26,23)(11,27,28,29)(13,30,16,33)(14,31,25,21)(18,34,19,36);(2,4)(5,7)(6,13)(8,17)(9,19)(10,20)(11,25)(12,30)(14,28)(15,16)(18,26)(21,27)(22,36)(23,34)(24,35)(29,31)(32,33)

Polycyclic presentation:

2^9|b^2=e,c^2=i,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,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, 325053);