#6450065

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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
prime radical	C₂×#3460	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3233 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305350, #72305359, #72305368
maximal normal subgroup of	#72305350, #72305359, #72305368
maximal quotient of	#72305551, #72305715

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6357261);