#383542

order	512
nilpotency class	3
isoclinism	isoclinic class of #30496
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1383 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)(5,6,11,15,12,8,10,16)(7,17,20,13,21,18,27,22)(9,14,25,32,26,23,28,31)(19,33,34,29,35,30,24,36);(5,7)(6,13)(8,22)(9,24)(10,27)(11,20)(12,21)(14,30)(15,18)(16,17)(19,25)(23,33)(26,34)(28,35)(29,31)(32,36);(2,4)(5,8)(6,12)(7,18)(9,23)(10,16)(11,15)(13,20)(14,26)(17,21)(19,33)(22,27)(24,29)(25,32)(28,31)(30,35)(34,36);(2,4)(5,9,10,25,12,26,11,28)(6,14,15,31,8,23,16,32)(7,19,20,24,21,35,27,34)(13,29,17,30,22,36,18,33)

Polycyclic presentation:

2^9|a^2=h,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 290738);