#7514153

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

3065 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)(7,19)(8,24)(9,28)(10,18)(11,17)(12,22)(13,27)(14,30)(15,29)(16,33)(20,31)(21,26)(23,34)(25,32)(35,36);(1,2)(3,4)(5,7,11,23)(6,12,17,25)(8,15,27,20)(9,13,30,24)(10,32,33,22)(14,36,28,26)(16,34,18,19)(21,29,35,31);(2,4)(5,8,11,27)(6,13,17,24)(7,20,23,15)(9,25,30,12)(10,26,33,36)(14,32,28,22)(16,21,18,35)(19,29,34,31);(2,4)(5,9)(6,14)(7,21)(8,25)(10,29)(11,30)(12,27)(13,32)(15,16)(17,28)(18,20)(19,36)(22,24)(23,35)(26,34)(31,33);(5,9)(6,15)(7,13)(8,25)(10,31)(11,30)(12,27)(14,16)(17,20)(18,28)(19,26)(21,32)(22,35)(23,24)(29,33)(34,36)

Polycyclic presentation:

2^9|b^2=i,c^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, 7421349);