#7142952

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#44892	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2072 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050148);