#7198973

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	#45271	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1923 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)(13,14,20,23,19,25,18,24)(15,26,17,21,28,16,27,22);(5,6)(7,10)(8,11)(9,12)(13,15,19,28)(14,21,25,22)(16,23,26,24)(17,20,27,18);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,16)(14,17)(15,24)(18,21)(19,26)(20,22)(23,28)(25,27);(5,8,10,9)(6,11,7,12)(13,17,18,15,19,27,20,28)(14,22,24,16,25,21,23,26);(5,9,10,8)(6,11,7,12)(13,18,19,20)(14,23,25,24)(15,27,28,17)(16,22,26,21)

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,e^c=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, 7106169);