#7545860

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1752 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=gi,e^2=f,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^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, 7453056);