#401066

order	512
nilpotency class	3
isoclinism	isoclinic class of #395163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=efi,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=fi,h^d=hi

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, 308262);