#361672

order	512
nilpotency class	3
isoclinism	isoclinic class of #23906
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62812	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3642 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268868);