#379510

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	#2487
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

902 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=eg,d^2=eh,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^b=fi,e^c=ei,g^c=gi,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, 286706);