#6542545

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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

3299 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)(13,14,18,19,17,22,16,21)(15,20,25,26,23,27,24,28);(5,6)(7,10)(8,9)(11,12)(14,19)(16,18)(20,26)(21,22)(24,25)(27,28);(2,4)(6,9)(10,12)(13,15)(14,20)(16,24)(17,23)(18,25)(19,26)(21,28)(22,27);(5,7)(6,10)(8,11)(9,12)(15,23)(20,27)(24,25)(26,28);(7,11)(10,12)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308427, #72308474, #72308501
maximal normal subgroup of	#72308427, #72308474, #72308501

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6449741);