#395426

order	512
nilpotency class	3
isoclinism	isoclinic class of #393982
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62852	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3601 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)(7,17)(8,22,24,14)(9,15)(10,13)(11,16)(12,19)(18,31,33,36)(20,28)(21,26)(23,30,25,29)(27,32,35,34);(1,2)(3,4)(5,7)(6,12)(8,18)(9,20)(10,19)(11,21)(13,26)(14,27)(15,17)(16,28)(22,35)(23,34)(24,33)(25,32)(29,36)(30,31);(2,4)(6,13)(9,10)(12,26)(14,29)(15,16)(17,28)(19,20)(22,30)(23,25)(27,36)(31,35)(32,34);(5,8)(6,14)(7,18)(9,25)(10,23)(11,24)(12,27)(13,29)(15,30)(16,22)(17,31)(19,34)(20,32)(21,33)(26,36)(28,35)

Polycyclic presentation:

2^9|a^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,e^d=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, 302622);