#7542038

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1774 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)(7,10)(8,12)(9,11)(13,14,19,22)(15,23,26,41)(16,28,30,21)(17,33,27,20)(18,36,35,24)(25,39,29,42)(31,38,34,37)(32,40,43,44);(1,2)(3,4)(5,7,9,12)(6,10,11,8)(13,15,18,27,19,26,35,17)(14,20,24,23,22,33,36,41)(16,29,32,34,30,25,43,31)(21,38,40,39,28,37,44,42);(5,8,9,10)(6,7,11,12)(13,16,19,30)(14,21,22,28)(15,25,26,29)(17,34,27,31)(18,32,35,43)(20,37,33,38)(23,42,41,39)(24,40,36,44);(2,4)(6,11)(7,12)(14,22)(15,26)(16,30)(17,27)(24,36)(32,43)(37,38)(39,42);(13,17,35,26,19,27,18,15)(14,23,36,20,22,41,24,33)(16,31,43,25,30,34,32,29)(21,39,44,38,28,42,40,37)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=ei

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, 7449234);