#7514004

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514000
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	#36212	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3344 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)(9,10,13,17)(11,18,20,15)(12,14,21,16)(19,23,24,22);(1,2)(3,4)(5,6)(7,8)(10,14)(12,21)(15,22)(16,17)(18,23)(19,24);(6,8)(9,11,13,20)(10,15,17,18)(12,19,21,24)(14,22,16,23);(2,4)(6,8);(10,14)(12,21)(15,22)(16,17)(18,23)(19,24)

Polycyclic presentation:

2^9|a^2=i,c^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,f^e=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282583, #72282753, #72284539
maximal normal subgroup of	#72282583, #72282753, #72284539

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421200);