#378901

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

930 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=e,d^2=eh,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=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, 286097);