#380428

order	512
nilpotency class	3
isoclinism	isoclinic class of #30056
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

969 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)(5,6,11,16,12,15,10,9)(7,17,21,13,22,19,20,30)(8,23,26,14,27,28,25,24)(18,31,34,36,35,29,33,32);(5,7)(6,13)(8,18)(9,19)(10,21)(11,20)(12,22)(14,31)(15,30)(16,17)(23,36)(24,29)(25,34)(26,33)(27,35)(28,32);(5,8,12,27)(6,14,15,24)(7,18,22,35)(9,28,16,23)(10,26,11,25)(13,29,30,31)(17,32,19,36)(20,34,21,33);(5,9,10,15,12,16,11,6)(7,19,21,30,22,17,20,13)(8,24,25,28,27,14,26,23)(18,29,34,32,35,31,33,36)

Polycyclic presentation:

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

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, 287624);