#6460019

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23393	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2371 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:

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

Polycyclic presentation:

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

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, 6367215);