#6454433

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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	#2847	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2084 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361629);