#6467701

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2032 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257264, #72257604, #72263838, …
maximal normal subgroup of	#72257264, #72257604, #72263838, …
maximal quotient of	#72264868, #72280713

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374897);