#395425

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

3445 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^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,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, 302621);