#7523231

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1894 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,d^2=f,e^2=f,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=cgi,d^b=dh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7430427);