#6432167

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,h^b=hi,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237210, #72237228, #72237249, …
maximal normal subgroup of	#72237210, #72237228, #72237249, …
maximal quotient of	#72238948

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339363);