#6448967

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
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	#23391	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2448 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284213
maximal normal subgroup of	#72284213

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356163);