#6441797

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	#35223	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2355 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)(7,9)(8,10)(11,12)(13,14,16,21,18,23,17,22)(15,20,25,28,26,27,24,19);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15)(14,19)(16,24)(17,25)(18,26)(20,22)(21,27)(23,28);(2,4)(6,10)(9,12)(13,15,16,25,18,26,17,24)(14,20,21,28,23,27,22,19);(2,4)(13,15,16,25,18,26,17,24)(14,20,21,28,23,27,22,19);(13,16,18,17)(14,21,23,22)(15,24,26,25)(19,27,28,20)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,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, 6348993);