#387629

order	512
nilpotency class	3
isoclinism	isoclinic class of #232138
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1086 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=efi,d^2=fg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^c=hi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286334, #72286356, #72286370, …
maximal normal subgroup of	#72286334, #72286356, #72286370, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294825);