#7514426

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	#35687	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2667 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)(7,11)(8,10)(9,12)(13,14,17,19)(15,23,26,36)(16,27,29,38)(18,31,30,22)(20,32,34,24)(21,33,35,28)(25,37,41,43)(39,40,44,42);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,19)(15,24)(16,28)(18,30)(20,36)(21,38)(23,34)(26,32)(27,35)(29,33)(37,43)(40,42);(13,15,17,26)(14,20,19,34)(16,25,29,41)(18,32,30,24)(21,37,35,43)(22,36,31,23)(27,39,38,44)(28,40,33,42);(2,4)(13,16,17,29)(14,21,19,35)(15,25,26,41)(18,33,30,28)(20,37,34,43)(22,38,31,27)(23,39,36,44)(24,40,32,42);(6,10)(9,11)(14,19)(16,29)(20,34)(22,31)(23,36)(25,41)(28,33)(40,42)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,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, 7421622);