#370442

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=efg,d^2=e,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, 277638);