#7198877

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996671
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	#24237	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1896 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=gi,d^2=fhi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7106073);