#7613764

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
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	#59550	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2014 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7520960);