#384417

order	512
nilpotency class	3
isoclinism	isoclinic class of #232128
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
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

961 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)(7,17,21,27)(8,22,26,14)(9,15)(10,13)(11,16)(12,19,29,20)(18,33)(23,28)(24,30,25,32)(31,35)(34,36);(1,2)(3,4)(5,7)(6,12)(8,23)(9,20)(10,19)(11,21)(13,17)(14,31)(15,27)(16,29)(18,24)(22,33)(25,35)(26,34)(28,32)(30,36);(2,4)(6,13,16,15)(12,27,29,17)(14,30,22,32)(28,33,36,31);(5,8,9,24,11,26,10,25)(6,14,15,32,16,22,13,30)(7,18,19,23,21,35,20,34)(12,28,17,31,29,36,27,33)

Polycyclic presentation:

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

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, 291613);