#358419

order	512
nilpotency class	3
isoclinism	isoclinic class of #188759
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

980 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=e,e^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,h^b=hi,d^c=di,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240851, #72252967, #72256869
maximal normal subgroup of	#72240851, #72252967, #72256869

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265615);