#370113

order	512
nilpotency class	3
isoclinism	isoclinic class of #24408
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1300 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)(7,17,20,12)(8,21)(9,16)(10,15)(11,14)(13,24)(18,34,35,29)(19,28,26,32)(22,33,36,27)(23,30)(25,31);(1,3,4,2)(5,7)(6,12)(8,22)(9,19)(10,26)(11,20)(13,29)(14,17)(15,28)(16,32)(18,25)(21,33)(23,36)(24,35)(27,30)(31,34);(5,8,10,25,11,23,9,24)(6,13,16,30,14,31,15,21)(7,18,19,22,20,35,26,36)(12,27,28,29,17,33,32,34);(5,7)(6,12)(8,18)(9,26)(10,19)(11,20)(13,27)(14,17)(15,32)(16,28)(21,34)(22,25)(23,35)(24,36)(29,30)(31,33)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277309);