#6459503

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438967
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#34737	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2556 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)(7,9)(8,10)(11,12)(13,14)(15,22)(16,21)(17,19)(18,26)(20,25)(23,27)(24,28);(2,4)(5,7)(6,9)(8,11)(10,12)(14,19)(20,27)(21,26)(22,28);(6,10)(7,11)(13,15,18,25,17,24,16,23)(14,20,21,22,19,27,26,28);(7,11)(9,12)(13,16,17,18)(14,21,19,26)(15,23,24,25)(20,22,27,28);(15,24)(20,27)(22,28)(23,25)

Polycyclic presentation:

2^9|a^2=fi,c^2=gh,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,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	#72312786, #72312897, #72312972
maximal normal subgroup of	#72312786, #72312897, #72312972

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366699);