#392431

order	512
nilpotency class	3
isoclinism	isoclinic class of #188764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂×#381
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1398 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)(7,12,19,27)(8,21,9,13)(10,15)(11,16)(14,23,29,22)(17,31)(18,25)(20,28,34,36)(24,30)(26,32)(33,35);(1,2,4,3)(5,7,10,19)(6,12,15,27)(8,17,9,18)(11,20,24,34)(13,25,21,31)(14,26,29,35)(16,28,30,36)(22,32,23,33);(5,8)(6,13)(7,17)(9,10)(11,22)(12,25)(14,30)(15,21)(16,29)(18,19)(20,32)(23,24)(26,36)(27,31)(28,35)(33,34);(1,3,4,2)(5,9,10,8)(6,14)(7,18,19,17)(11,23,24,22)(12,26)(13,16)(15,29)(20,33,34,32)(21,30)(25,28)(27,35)(31,36)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,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	#72279790, #72279855, #72279937, …
maximal normal subgroup of	#72279790, #72279855, #72279937, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299627);