#397915

order	512
nilpotency class	3
isoclinism	isoclinic class of #394358
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23445	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1907 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251058, #72251082, #72251089
maximal normal subgroup of	#72251058, #72251082, #72251089
maximal quotient of	#72314514

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305111);