#203899

order	512
nilpotency class	3
isoclinism	isoclinic class of #19552
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

674 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fh,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,e^b=eh,d^c=dhi

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, 111095);