#7562154

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	#44884	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2221 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)(14,20)(15,22)(16,26)(17,19)(21,24)(25,27);(1,2)(3,4)(6,10)(11,12)(13,14)(15,24)(16,25)(17,23)(18,20)(19,28)(21,27)(22,26);(5,7,9,8)(6,11,10,12)(13,15,17,27,18,25,19,22)(14,21,28,24,20,26,23,16);(2,4)(5,8,9,7)(6,12,10,11)(13,16,19,26,18,24,17,21)(14,22,23,25,20,27,28,15);(6,10)(11,12)(13,17,18,19)(14,23,20,28)(15,22,25,27)(16,21,24,26)

Polycyclic presentation:

2^9|c^2=fg,d^2=fgi,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=cgi,d^b=dh,g^b=gi,h^b=hi,e^c=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, 7469350);