#209535

order	512
nilpotency class	3
isoclinism	isoclinic class of #18446
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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

1023 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)(7,16)(8,13)(9,14)(10,19)(11,15)(12,17)(18,20)(21,22,25,28)(23,26,30,33)(24,27,31,34)(29,35,36,32);(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)(24,31)(26,33)(29,36)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=ghi,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, 116731);