#395424

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

3603 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:

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

Polycyclic presentation:

2^9|a^2=ef,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, 302620);