#7561615

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

2145 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)(7,11)(8,12)(9,10);(2,3)(6,10)(11,12)(13,14,18,22)(15,19,24,34)(16,26,28,38)(17,30,32,21)(20,31,37,29)(23,39,27,44)(25,42,41,35)(33,43,36,40);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,15)(14,19)(16,27)(17,25)(18,24)(20,36)(21,35)(22,34)(23,28)(26,44)(29,43)(30,42)(31,40)(32,41)(33,37)(38,39);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,16)(14,20)(15,23)(17,31)(18,28)(19,33)(21,38)(22,37)(24,27)(25,43)(26,30)(29,32)(34,36)(35,44)(39,42)(40,41);(6,10)(11,12)(15,24)(16,28)(19,34)(20,37)(25,41)(26,38)(29,31)(35,42)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=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, 7468811);