#7504030

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1823 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6)(7,8)(9,11)(10,12)(13,14,18,24,19,23,34,35)(15,20,26,21,16,27,29,38)(17,30,31,22,32,40,33,39)(25,37,41,36,28,44,42,43);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,19,16)(14,20,23,27)(17,25,32,28)(18,26,34,29)(21,35,38,24)(22,36,39,43)(30,37,40,44)(31,41,33,42);(2,4)(13,16,19,15)(14,21,23,38)(17,28,32,25)(18,26,34,29)(20,35,27,24)(22,37,39,44)(30,36,40,43)(31,41,33,42);(6,11)(7,12)(13,17,19,32)(14,22,23,39)(15,25,16,28)(18,33,34,31)(20,36,27,43)(21,37,38,44)(24,30,35,40)(26,42,29,41);(5,8,9,10)(6,7,11,12)(14,23)(15,16)(22,39)(24,35)(25,28)(26,29)(30,40)(41,42)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi

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, 7411226);