#209529

order	512
nilpotency class	3
isoclinism	isoclinic class of #18446
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1013 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)(7,14)(8,12)(9,17)(10,13)(11,15)(16,19)(18,20)(21,22)(23,26)(24,27)(25,28)(29,35)(30,33)(31,34)(32,36);(5,7,10,16)(6,11,13,20)(8,15,9,18)(12,19,17,14)(21,23)(22,26)(24,29)(25,30)(27,32)(28,33)(31,36)(34,35);(5,6)(7,11)(8,17)(9,12)(10,13)(14,15)(16,20)(18,19)(21,24)(22,27)(23,29)(25,31)(26,32)(28,34)(30,36)(33,35);(1,3,4,2)(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14)(22,28)(24,31)(26,33)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116725);