#6470103

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470092
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4465 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)(7,9)(8,10)(11,12)(13,14)(15,25)(16,28)(17,22)(18,24)(19,23)(20,26)(21,27);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15,19,27)(14,20,23,28)(16,18,26,17)(21,22,25,24);(7,11)(9,12)(13,16,19,26)(14,21,23,25)(15,17,27,18)(20,24,28,22);(7,11)(9,12)(13,17,19,18)(14,22,23,24)(15,16,27,26)(20,25,28,21);(6,10)(9,12)(14,23)(20,28)(21,25)(22,24)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282750, #72282916
maximal normal subgroup of	#72282750, #72282916

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6377299);