#357913

order	512
nilpotency class	3
isoclinism	isoclinic class of #2048
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#394
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2498 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237178, #72237184, #72237192, …
maximal normal subgroup of	#72237178, #72237184, #72237192, …
maximal quotient of	#72239076

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265109);