#6591538

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4429 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:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284628, #72285337
maximal normal subgroup of	#72284628, #72285337

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6498734);