#7199174

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1834 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)(13,14,19,17,20,21,18,28)(15,22,25,23,26,27,24,16);(5,6)(7,10)(8,11)(9,12)(13,15,20,26)(14,16,21,23)(17,27,28,22)(18,25,19,24);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,16)(14,15)(17,25)(18,27)(19,22)(20,23)(21,26)(24,28);(5,8,10,9)(6,11,7,12)(13,17,18,14,20,28,19,21)(15,23,24,22,26,16,25,27);(5,9,10,8)(6,11,7,12)(13,18,20,19)(14,17,21,28)(15,24,26,25)(16,22,23,27)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=fh,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=ei

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, 7106370);