#112295

order	512
nilpotency class	3
isoclinism	isoclinic class of #97710
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

847 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dh,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, 19491);