#392327

order	512
nilpotency class	3
isoclinism	isoclinic class of #188764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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

1302 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)(7,11,17,26)(8,19,20,13)(9,15)(10,14)(12,21,27,33)(16,29)(18,25,32,36)(22,28)(23,34)(24,30)(31,35);(1,2,4,3)(5,7,10,18,9,17,22,32)(6,11,14,25,15,26,28,36)(8,16,21,31,20,30,33,34)(12,23,19,29,27,35,13,24);(1,3,4,2)(5,8,9,20)(6,12)(7,16,17,30)(10,21,22,33)(11,23)(13,28)(14,19)(15,27)(18,31,32,34)(24,36)(25,29)(26,35);(5,8)(6,13)(7,16)(9,20)(10,21)(11,24)(12,14)(15,19)(17,30)(18,31)(22,33)(23,25)(26,29)(27,28)(32,34)(35,36)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,h^b=hi,d^c=di,h^c=hi,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250272, #72250285, #72250289, …
maximal normal subgroup of	#72250272, #72250285, #72250289, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299523);