#7514290

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
prime radical	#36213	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3260 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)(9,10,14,19)(11,17,12,16)(13,22,23,18)(15,21,24,20);(1,2)(3,4)(5,6)(7,8)(9,11,14,12)(10,15,19,24)(13,21,23,20)(16,22,17,18);(6,8)(9,11,14,12)(10,16,19,17)(13,20,23,21)(15,22,24,18);(2,4)(6,8)(9,12,14,11)(10,17,19,16)(13,21,23,20)(15,18,24,22);(10,18)(13,23)(15,16)(17,24)(19,22)(20,21)

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=efi,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	#72315616, #72315668, #72315989
maximal normal subgroup of	#72315616, #72315668, #72315989

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421486);