#383765

order	512
nilpotency class	3
isoclinism	isoclinic class of #31418
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1344 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ei,d^2=e,g^2=i,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,g^b=gi,h^b=hi,f^c=fi,g^c=gi,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, 290961);