#7522437

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	#45332	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2163 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=fi,e^2=f,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, 7429633);