#370443

order	512
nilpotency class	3
isoclinism	isoclinic class of #27987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
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

724 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)(5,6,10,14,11,15,9,26)(7,16,20,17,21,28,19,12)(8,22,24,29,25,35,23,13)(18,31,33,36,34,27,32,30);(1,3,4,2)(5,7,10,20,11,21,9,19)(6,12,26,28,15,17,14,16)(8,18,24,33,25,34,23,32)(13,27,35,36,29,31,22,30);(5,6)(7,17,21,12)(8,13)(9,14)(10,26)(11,15)(16,19,28,20)(18,31,34,27)(22,23)(24,35)(25,29)(30,32,36,33);(5,8,10,24,11,25,9,23)(6,13,26,35,15,29,14,22)(7,18,20,33,21,34,19,32)(12,27,28,36,17,31,16,30)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=efg,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,h^c=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, 277639);