#384441

order	512
nilpotency class	3
isoclinism	isoclinic class of #232138
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

962 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)(7,18)(8,13,25,14)(9,26,27,33)(10,16)(11,17)(12,21)(15,24)(19,34,35,32)(20,36,23,29)(22,30)(28,31);(1,2)(3,4)(5,7,11,22)(6,12,17,31)(8,23,25,20)(9,19,27,35)(10,28,24,21)(13,32,14,34)(15,30,16,18)(26,29,33,36);(2,4)(5,8,24,27,11,25,10,9)(6,13,15,33,17,14,16,26)(7,19,28,20,22,35,21,23)(12,29,18,34,31,36,30,32);(5,9,10,25,11,27,24,8)(6,14,15,26,17,13,16,33)(7,20,21,19,22,23,28,35)(12,29,18,34,31,36,30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ef,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^c=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, 291637);