#6467638

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	#25788	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2395 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)(7,10)(8,9)(11,12)(13,14,18,23,20,24,19,16)(15,25,28,21,26,27,17,22);(2,4)(13,15)(14,21)(16,27)(17,18)(19,28)(20,26)(22,24)(23,25);(6,9)(10,12)(13,16,18,14,20,23,19,24)(15,22,28,25,26,21,17,27);(13,17,18,15,20,28,19,26)(14,22,23,25,24,21,16,27);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,23,24,16)(15,17,26,28)(21,25,22,27)

Polycyclic presentation:

2^9|a^2=fi,c^2=hi,d^2=hi,e^2=g,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^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, 6374834);