#6439657

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	#23421	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2304 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307329, #72307403, #72307478
maximal normal subgroup of	#72307329, #72307403, #72307478

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346853);