#357879

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

2602 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312497, #72312522, #72312574
maximal normal subgroup of	#72312497, #72312522, #72312574

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265075);