#7551174

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	C₂×#3271	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2103 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)(7,11)(8,9)(10,12)(13,14,20,25)(15,26,18,28)(16,27,23,21)(17,24,19,22);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,27,20,18,19,21)(14,16,22,28,25,23,24,26);(6,9)(10,11)(13,16,20,23)(14,21,25,27)(15,24,18,22)(17,28,19,26);(2,4)(5,7,8,12)(6,10,9,11)(13,17,20,19)(14,22,25,24)(15,21,18,27)(16,26,23,28);(13,18,19,27,20,15,17,21)(14,23,24,28,25,16,22,26)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,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, 7458370);