#383759

order	512
nilpotency class	3
isoclinism	isoclinic class of #213760
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1346 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,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, 290955);