#6606708

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#37864	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1906 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)(13,14,19,23,18,15,17,21)(16,20,26,28,24,22,25,27);(5,6,8,9)(7,10,11,12)(13,15,18,14)(16,22,24,20)(17,21,19,23)(25,27,26,28);(1,2,3,4)(6,9)(7,11)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28);(5,7)(6,10)(8,11)(9,12)(16,24)(20,22)(25,26)(27,28);(6,9)(7,11)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307480, #72307677, #72308001
maximal normal subgroup of	#72307480, #72307677, #72308001

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6513904);