#212899

order	512
nilpotency class	3
isoclinism	isoclinic class of #19713
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

722 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 120095);