#7545920

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

1743 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)(15,23)(16,28)(17,33)(18,36)(19,22)(20,27)(21,30)(24,35)(25,39)(26,41)(29,42)(31,38)(32,40)(34,37)(43,44);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(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);(5,8,9,12)(6,10,7,11)(13,16)(14,21)(15,25)(17,34)(18,32)(19,30)(20,37)(22,28)(23,42)(24,40)(26,29)(27,31)(33,38)(35,43)(36,44)(39,41);(2,4)(6,7)(10,11)(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=gi,b^2=fgi,c^2=gi,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, 7453116);