#366001

order	512
nilpotency class	3
isoclinism	isoclinic class of #24987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

1095 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=i,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^b=hi,g^c=gi,h^c=hi,e^d=ei,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, 273197);