#357446

order	512
nilpotency class	3
isoclinism	isoclinic class of #9109
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#3449	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4406 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72236660, #72249684, #72249737, …
maximal normal subgroup of	#72236660, #72249684, #72249737, …
maximal quotient of	#72236686, #72280897, #72303897

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264642);