#115757

order	512
nilpotency class	3
isoclinism	isoclinic class of #100227
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 22953);