#6438476

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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×#3045	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2364 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)(5,6,8,9)(7,10,11,12)(13,14)(15,22)(16,27)(17,21)(18,20)(19,26)(23,28)(24,25);(6,9)(10,12)(13,15,17,26)(14,19,21,22)(16,25,18,23)(20,24,27,28);(2,4)(5,7,8,11)(6,10,9,12)(15,23)(16,18)(19,24)(20,27)(22,28)(25,26);(13,14)(15,24)(16,20)(17,21)(18,27)(19,23)(22,25)(26,28);(13,16,17,18)(14,20,21,27)(15,25,26,23)(19,24,22,28)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307281, #72307448, #72307596
maximal normal subgroup of	#72307281, #72307448, #72307596

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345672);