#394111

order	512
nilpotency class	3
isoclinism	isoclinic class of #394046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

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

4043 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72278124, #72314443, #72314468
maximal normal subgroup of	#72278124, #72314443, #72314468
maximal quotient of	#72278137, #72314500

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301307);