#7514161

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514160
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

3055 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)(5,6,10,15)(7,12,19,29)(8,20,24,32)(9,11,25,14)(13,23,31,22)(16,28,18,26)(17,34,21,36)(27,35,30,33);(1,2)(3,4)(6,11)(7,16)(9,25)(13,20)(14,15)(17,33)(18,19)(21,35)(22,23)(26,28)(31,32)(34,36);(2,4);(2,4)(5,7)(6,12)(8,21)(9,18)(10,19)(11,26)(13,30)(14,28)(15,29)(16,25)(17,24)(20,36)(22,35)(23,33)(27,31)(32,34);(5,8)(6,13)(7,17)(9,23)(10,24)(11,20)(12,27)(14,32)(15,31)(16,33)(18,35)(19,21)(22,25)(26,34)(28,36)(29,30)

Polycyclic presentation:

2^9|a^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,e^d=ei,h^d=hi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421357);