#6467149

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437412
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	#24214	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2194 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)(7,10)(8,9)(11,12)(13,14)(15,22)(16,21)(17,25)(18,20)(19,23)(24,28)(26,27);(2,3)(13,15)(14,19)(16,23)(17,26)(18,24)(20,27)(21,28)(22,25);(1,2,4,3)(6,9)(10,12)(14,20)(19,27)(21,25)(22,28);(1,3,4,2)(13,16,18,17)(14,21,20,25)(15,23,24,26)(19,28,27,22);(5,7,8,11)(6,10,9,12)(14,20)(19,27)(21,25)(22,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282430, #72308284, #72308800
maximal normal subgroup of	#72282430, #72308284, #72308800

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374345);