#449500

order	512
nilpotency class	3
isoclinism	isoclinic class of #417534
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3237 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)(6,11)(9,21)(12,27)(13,26)(14,22)(16,23)(18,30)(19,24)(25,29)(28,35)(31,36)(32,34);(5,6)(7,12)(8,18)(9,22)(10,14)(11,21)(13,19)(15,31)(16,25)(17,29)(20,26)(23,27)(24,30)(28,32)(33,35)(34,36);(5,7,10,17)(6,12,14,29)(8,15,20,33)(9,23,21,16)(11,25,22,27)(13,28,30,36)(18,35,26,31)(19,34,24,32);(2,4)(5,8)(6,13)(7,15)(9,24)(10,20)(11,26)(12,28)(14,30)(16,34)(17,33)(18,22)(19,21)(23,32)(25,31)(27,35)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263003, #72263055, #72263056, …
maximal normal subgroup of	#72263003, #72263055, #72263056, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356696);