#7503811

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

2198 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)(13,14,17,33,18,24,19,23)(15,25,34,38,29,20,28,37)(16,30,35,21,32,39,31,22)(26,41,42,40,27,44,43,36);(1,2)(3,4)(5,7,9,11)(6,10,12,8)(13,15)(14,20)(16,26)(17,34)(18,29)(19,28)(21,36)(22,40)(23,38)(24,25)(27,32)(30,44)(31,43)(33,37)(35,42)(39,41);(2,4)(5,8,9,10)(6,11,12,7)(13,16)(14,21)(15,26)(17,31)(18,32)(19,35)(20,36)(22,24)(23,39)(25,40)(27,29)(28,42)(30,33)(34,43)(37,44)(38,41);(6,12)(7,11)(13,16)(14,22)(15,27)(17,31)(18,32)(19,35)(20,36)(21,24)(23,30)(25,40)(26,29)(28,43)(33,39)(34,42)(37,44)(38,41);(13,17,18,19)(14,23,24,33)(15,28,29,34)(16,31,32,35)(20,37,25,38)(21,39,22,30)(26,42,27,43)(36,44,40,41)

Polycyclic presentation:

2^9|a^2=fg,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, 7411007);