#395469

order	512
nilpotency class	3
isoclinism	isoclinic class of #394046
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

3602 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)(7,16,18,11)(8,20,21,12)(9,15)(10,14)(13,22,29,24)(17,32)(19,31,23,28)(25,30)(26,34)(27,35)(33,36);(1,2)(3,4)(5,7)(6,11)(8,17)(9,23)(10,18)(12,26)(13,27)(14,16)(15,31)(19,25)(20,32)(21,34)(22,36)(24,35)(28,30)(29,33);(2,4)(5,8)(6,12)(7,17)(9,22)(10,21)(11,26)(13,15)(14,20)(16,32)(18,34)(19,35)(23,36)(24,25)(27,31)(28,33)(29,30);(5,8)(6,13)(7,17)(9,24)(10,21)(11,27)(12,15)(14,29)(16,33)(18,34)(19,36)(20,30)(22,25)(23,35)(26,31)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282010
maximal normal subgroup of	#72282010
maximal quotient of	#72282064

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302665);