#6457670

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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	#35272	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2290 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)(7,11)(8,12)(9,10)(13,14,19,24,20,26,18,25)(15,27,28,16,22,17,23,21);(6,10)(11,12)(13,15)(14,21)(16,26)(17,24)(18,28)(19,23)(20,22)(25,27);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,16,19,17,20,21,18,27)(14,22,24,23,26,15,25,28);(5,8,9,7)(6,12,10,11)(13,17,19,21,20,27,18,16)(14,23,24,15,26,28,25,22);(13,18,20,19)(14,24,26,25)(15,28,22,23)(16,27,21,17)

Polycyclic presentation:

2^9|a^2=hi,c^2=fg,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, 6364866);