#115871

order	512
nilpotency class	3
isoclinism	isoclinic class of #99803
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	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

710 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)(7,13)(8,10)(9,17)(11,15)(12,16)(14,20)(18,19)(21,22)(23,30,33,36)(24,28)(25,34)(26,29)(27,32,35,31);(1,2,3,4)(6,10)(11,19)(12,17)(13,20)(23,31)(24,25)(27,30)(28,34)(32,33)(35,36);(5,7,8,14)(6,11,10,19)(9,18,16,15)(12,20,17,13)(21,23)(22,27)(24,31)(25,32)(26,33)(28,36)(29,35)(30,34)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23067);