#6456554

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438967
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	#34976	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2042 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=f,d^2=f,e^2=gi,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,d^c=di,e^c=ei,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, 6363750);