#388734

order	512
nilpotency class	3
isoclinism	isoclinic class of #30434
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

875 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=h,d^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,g^b=gi,f^c=fi,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, 295930);