#7557542

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1761 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=i,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7464738);