#6470106

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

4458 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)(15,21)(16,20)(17,22)(18,23)(19,24)(25,28)(26,27);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,18)(17,25)(19,26)(21,22)(23,27)(24,28);(7,11)(9,12)(13,16)(14,21)(15,17)(18,26)(19,25)(20,23)(22,28)(24,27);(7,11)(9,12)(13,17,19,18)(14,22,24,23)(15,25,26,16)(20,21,28,27);(6,10)(9,12)(13,17,19,18)(14,23,24,22)(15,25,26,16)(20,27,28,21)

Polycyclic presentation:

2^9|d^2=i,e^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	#72282127, #72282154, #72282979
maximal normal subgroup of	#72282127, #72282154, #72282979

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6377302);