#7514159

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514148
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

3068 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)(5,6,12,17)(7,19,23,32)(8,24,28,14)(9,29,26,15)(10,22,13,33)(11,18,34,16)(20,31,25,30)(21,36,35,27);(1,2)(3,4)(5,7,12,23)(6,13,17,10)(8,25,28,20)(9,14,26,24)(11,33,34,22)(15,27,29,36)(16,32,18,19)(21,30,35,31);(2,4)(5,8,12,28)(6,14,17,24)(7,20,23,25)(9,10,26,13)(11,27,34,36)(15,22,29,33)(16,21,18,35)(19,30,32,31);(2,4)(5,9,12,26)(6,15,17,29)(7,21,23,35)(8,10,28,13)(11,30,34,31)(14,22,24,33)(16,20,18,25)(19,27,32,36);(5,10)(6,7)(8,26)(9,28)(11,19)(12,13)(14,20)(15,21)(16,33)(17,23)(18,22)(24,25)(27,30)(29,35)(31,36)(32,34)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,f^b=fi,h^b=hi,h^d=hi,f^e=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421355);