#7624941

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624824
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1171 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,28)(19,30,31,25)(20,33,37,47)(21,27,24,44)(23,43,42,48)(26,45,40,38)(29,41,34,35)(32,36,46,39);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,22,31)(18,25,28,30)(20,34,37,29)(21,40,24,26)(23,32,42,46)(27,45,44,38)(33,41,47,35)(36,43,39,48);(17,20,35,25,24,39,42,45,22,37,41,30,21,36,23,38)(18,26,43,46,27,31,34,47,28,40,48,32,44,19,29,33);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,21,22,24)(18,27,28,44)(19,26,31,40)(20,36,37,39)(23,41,42,35)(25,38,30,45)(29,43,34,48)(32,33,46,47);(2,8)(6,9)(7,16)(14,15)(18,28)(19,31)(26,40)(27,44)(29,34)(32,46)(33,47)(43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gh,d^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532137);