#6444997

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 ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2286 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)(16,20)(17,26)(18,21)(19,25)(23,28)(24,27);(1,2)(3,4)(6,9)(10,12)(15,23)(19,27)(22,28)(24,25);(2,4)(13,15,17,25,18,23,16,24)(14,19,20,22,21,27,26,28);(2,4)(13,16,18,17)(14,20,21,26)(15,24,23,25)(19,22,27,28);(5,7,8,11)(6,10,9,12)(15,23)(19,27)(22,28)(24,25)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250181, #72256988, #72257687, …
maximal normal subgroup of	#72250181, #72256988, #72257687, …
maximal quotient of	#72264668, #72265429, #72308269

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352193);