#209500

order	512
nilpotency class	3
isoclinism	isoclinic class of #18414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1021 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 116696);