#6460022

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	#35564	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2297 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gh,d^2=f,e^2=i,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, 6367218);