#7518641

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514000
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24370	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2121 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(5,6)(7,11)(8,9)(10,12)(13,14,17,15)(16,24,26,25)(18,22,19,21)(20,28,23,27);(1,2)(3,4)(6,9)(10,11)(13,15)(14,17)(16,25)(18,21)(19,22)(20,28)(23,27)(24,26);(13,16,17,26)(14,20,15,23)(18,28,19,27)(21,25,22,24);(2,4)(14,21)(15,22)(18,19)(20,25)(23,24)(27,28);(5,7,8,12)(6,10,9,11)(14,15)(20,23)(21,22)(24,25)

Polycyclic presentation:

2^9|a^2=i,c^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,g^d=gi,h^d=hi

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, 7425837);