#7527805

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	#36018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2012 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)(5,6,8,10)(7,9,11,12)(13,14,18,22)(15,19,24,33)(16,20,28,37)(17,21,31,40)(23,32,41,43)(25,34,29,38)(26,35,42,44)(27,36,30,39);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,19)(16,26)(17,29)(18,24)(20,35)(21,38)(22,33)(23,27)(25,31)(28,42)(30,41)(32,36)(34,40)(37,44)(39,43);(6,10)(7,11)(13,15)(14,19)(16,23)(17,25)(18,24)(20,32)(21,34)(22,33)(26,27)(28,41)(29,31)(30,42)(35,36)(37,43)(38,40)(39,44);(6,10)(9,12)(13,16,18,28)(14,20,22,37)(15,23,24,41)(17,30,31,27)(19,32,33,43)(21,39,40,36)(25,42,29,26)(34,44,38,35);(1,3,4,2)(15,24)(19,33)(23,41)(25,29)(26,42)(32,43)(34,38)(35,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,d^2=i,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, 7435001);