#7503812

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=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, 7411008);