#6459449

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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	C₂×#3013	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2785 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)(7,9)(8,10)(11,12)(13,14,18,23)(15,21,25,20)(16,22,17,27)(19,26,28,24);(5,7)(6,9)(8,11)(10,12)(13,15,18,25)(14,19,23,28)(16,24,17,26)(20,27,21,22);(1,2,3,4)(6,10)(7,11)(13,15)(14,20)(16,26)(17,24)(18,25)(19,22)(21,23)(27,28);(7,11)(9,12)(13,15)(14,21)(16,26)(17,24)(18,25)(19,27)(20,23)(22,28);(13,16,18,17)(14,22,23,27)(15,24,25,26)(19,20,28,21)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,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, 6366645);