#7514226

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514196
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3375 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)(9,10,14,20)(11,17,12,16)(13,18,23,19)(15,22,24,21);(1,2)(3,4)(5,6)(7,8)(9,11,14,12)(10,15,20,24)(13,22,23,21)(16,18,17,19);(6,8)(9,11,14,12)(10,16,20,17)(13,21,23,22)(15,18,24,19);(2,4)(6,8)(9,12,14,11)(10,17,20,16)(13,22,23,21)(15,19,24,18);(10,18)(13,23)(15,17)(16,24)(19,20)(21,22)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315640, #72315693, #72315989
maximal normal subgroup of	#72315640, #72315693, #72315989

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421422);