#7142937

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	#45263	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2067 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)(14,18)(15,19)(16,23)(21,27)(22,25)(26,28);(1,2)(3,4)(6,9)(7,11)(13,14,17,18)(15,19,22,25)(16,24,23,20)(21,27,28,26);(5,6)(7,10)(8,9)(11,12)(13,15,17,22)(14,19,18,25)(16,21,23,28)(20,26,24,27);(2,4)(6,9)(10,12)(13,16,17,23)(14,20,18,24)(15,21,22,28)(19,26,25,27);(5,7,8,11)(6,10,9,12)(16,23)(20,24)(21,28)(26,27)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,e^2=h,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, 7050133);