#6459772

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	C₂×#3008	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3236 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)(7,9)(8,10)(11,12)(13,14)(15,19)(16,20)(17,23)(18,21)(22,27)(24,25)(26,28);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,14,19,23,18,21,17,15)(16,24,27,22,25,20,26,28);(6,10)(7,11)(13,15,19,14,18,23,17,21)(16,22,27,20,25,28,26,24);(7,11)(9,12)(13,16)(14,20)(15,22)(17,26)(18,25)(19,27)(21,24)(23,28);(16,25)(20,24)(22,28)(26,27)

Polycyclic presentation:

2^9|b^2=fi,c^2=gh,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,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, 6366968);