#361673

order	512
nilpotency class	3
isoclinism	isoclinic class of #23912
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

3628 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268869);