#379816

order	512
nilpotency class	3
isoclinism	isoclinic class of #214688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2106 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)(7,16,19,11)(8,13)(9,15)(10,14)(12,18,27,22)(17,33,35,25)(20,30)(21,29)(23,32)(24,31)(26,36,28,34);(1,2)(3,4)(5,7)(6,11)(8,17)(9,18)(10,19)(12,15)(13,25)(14,16)(20,34)(21,35)(22,24)(23,36)(26,32)(27,31)(28,30)(29,33);(2,4)(5,7)(6,12)(8,17)(9,22)(10,19)(11,15)(13,28)(14,27)(16,31)(18,24)(20,36)(21,35)(23,34)(25,30)(26,29)(32,33);(5,8,10,21)(6,13,14,29)(7,17,19,35)(9,23,24,20)(11,25,16,33)(12,26,27,28)(15,32,31,30)(18,36,22,34)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287012);