#6629292

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628842
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4470 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,e^c=ei,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282340, #72308789, #72308822
maximal normal subgroup of	#72282340, #72308789, #72308822

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536488);