#7003301

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#44869	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1972 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6910497);