#6441752

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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	#35252	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2358 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)(13,14,18,26,20,15,19,23)(16,24,27,25,28,21,17,22);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15,20,14)(16,22,28,25)(17,24,27,21)(18,26,19,23);(2,4)(6,10)(9,12)(13,16,19,17,20,28,18,27)(14,21,23,25,15,24,26,22);(2,4)(13,17,19,28,20,27,18,16)(14,22,23,21,15,25,26,24);(13,18,20,19)(14,23,15,26)(16,27,28,17)(21,25,24,22)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348948);