#6441780

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

2388 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)(7,9)(8,10)(11,12)(13,14,17,22,19,21,18,16)(15,23,25,24,27,28,26,20);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15,19,27)(14,20,21,24)(16,23,22,28)(17,26,18,25);(2,4)(6,10)(9,12)(13,16,18,21,19,22,17,14)(15,24,26,23,27,20,25,28);(2,4)(13,14,17,22,19,21,18,16)(15,23,25,24,27,28,26,20);(14,21)(16,22)(20,24)(23,28)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307319, #72307559
maximal normal subgroup of	#72307319, #72307559

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348976);