#7515111

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	#25534	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1976 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)(7,10)(8,9)(11,12)(13,14,20,24)(15,21,29,40)(16,30,33,35)(17,32,22,37)(18,23,19,36)(25,41,43,34)(26,31,38,44)(27,39,28,42);(1,2)(3,4)(6,9)(10,12)(13,15,20,29)(14,21,24,40)(16,31,33,44)(17,34,22,41)(18,28,19,27)(23,42,36,39)(25,37,43,32)(26,30,38,35);(5,7,8,11)(6,10,9,12)(13,16)(14,22)(15,25)(17,24)(18,32)(19,37)(20,33)(21,38)(23,35)(26,40)(27,31)(28,44)(29,43)(30,36)(34,39)(41,42);(2,4)(13,17,20,22)(14,16,24,33)(15,26,29,38)(18,35,19,30)(21,25,40,43)(23,37,36,32)(27,34,28,41)(31,39,44,42);(6,9)(10,12)(13,18,20,19)(14,23,24,36)(15,27,29,28)(16,32,33,37)(17,30,22,35)(21,39,40,42)(25,31,43,44)(26,41,38,34)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=i,e^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,e^b=ei,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, 7422307);