#357651

order	512
nilpotency class	3
isoclinism	isoclinic class of #9103
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2730 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237646, #72237654, #72237671, …
maximal normal subgroup of	#72237646, #72237654, #72237671, …
maximal quotient of	#72314612

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264847);