#368556

order	512
nilpotency class	3
isoclinism	isoclinic class of #27987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1151 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=eg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,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, 275752);