#7514059

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#36082	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(5,6)(7,8)(9,10,15,21)(11,18,13,24)(12,20,14,17)(16,23,19,22);(1,2)(3,4)(5,6)(7,8)(9,11,15,13)(10,16,21,19)(12,22,14,23)(17,18,20,24);(6,8)(9,12,15,14)(10,17,21,20)(11,22,13,23)(16,18,19,24);(2,4)(6,8)(9,11,15,13)(10,18,21,24)(12,23,14,22)(16,20,19,17);(9,13,15,11)(10,19,21,16)(12,23,14,22)(17,24,20,18)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,f^a=fi,c^b=cgi,d^b=dh,f^b=fi,d^c=di,f^e=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421255);