#418825

order	512
nilpotency class	3
isoclinism	isoclinic class of #417661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60369	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1992 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:

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

Polycyclic presentation:

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

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, 326021);