#115808

order	512
nilpotency class	3
isoclinism	isoclinic class of #100051
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

576 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,e^2=hi,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,f^b=fh,e^c=ehi,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23004);