#6465514

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442590
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2502 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320498, #72320602, #72320791
maximal normal subgroup of	#72320498, #72320602, #72320791
maximal quotient of	#72322798

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372710);