#359022

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	#23438	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1850 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266218);