#365329

order	512
nilpotency class	3
isoclinism	isoclinic class of #24808
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

952 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)(5,6,10,14,11,16,9,15)(7,17,21,28,22,18,20,12)(8,23,25,30,26,29,24,13)(19,33,35,32,36,27,34,31);(1,3,4,2)(5,7,10,21,11,22,9,20)(6,12,15,18,16,28,14,17)(8,19,25,35,26,36,24,34)(13,27,29,32,30,33,23,31);(5,6)(7,18)(8,13)(9,14)(10,15)(11,16)(12,20)(17,22)(19,32)(21,28)(23,24)(25,29)(26,30)(27,34)(31,36)(33,35);(5,8,10,25,11,26,9,24)(6,13,15,29,16,30,14,23)(7,19,21,35,22,36,20,34)(12,27,18,32,28,33,17,31)

Polycyclic presentation:

2^9|a^2=h,b^2=efh,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,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, 272525);