#7514103

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

3267 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)(9,10,13,16)(11,18,19,15)(12,21,23,17)(14,22,24,20);(1,2)(3,4)(5,6)(7,8)(9,11,13,19)(10,14,16,24)(12,22,23,20)(15,21,18,17);(6,8)(9,11)(10,15)(12,20)(13,19)(14,17)(16,18)(21,24)(22,23);(2,4)(6,8)(10,16)(11,19)(17,21)(20,22);(10,17)(12,23)(14,15)(16,21)(18,24)(20,22)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315640, #72315707, #72315800
maximal normal subgroup of	#72315640, #72315707, #72315800

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421299);