#6437912

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437602
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3585	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317306
maximal normal subgroup of	#72317306

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345108);