#6443330

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437963
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62812	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311332, #72311366, #72311376, …
maximal normal subgroup of	#72311332, #72311366, #72311376, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350526);