#395427

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	#62859	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3443 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=i,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, 302623);