#7520157

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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1849 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)(8,12)(9,10)(13,14,18,22)(15,24,25,17)(16,23,19,21)(20,28,26,27);(1,2)(3,4)(6,10)(11,12)(13,15,18,25)(14,17,22,24)(16,26,19,20)(21,27,23,28);(5,7,9,8)(6,11,10,12)(13,16)(15,25)(17,27)(18,19)(21,23)(24,28);(2,4)(13,17,18,24)(14,20,22,26)(15,23,25,21)(16,27,19,28);(5,8,9,7)(6,11,10,12)(13,16,18,19)(14,21,22,23)(15,26,25,20)(17,27,24,28)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=i,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, 7427353);