#6483167

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2179 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)(7,12)(8,10)(9,11)(13,14,17,16)(15,21,22,28)(18,25,26,20)(19,27,23,24);(2,4)(7,8)(10,12)(14,16)(15,22)(20,25)(24,27);(5,7,9,8)(6,10,11,12)(13,15,18,24,17,22,26,27)(14,19,20,21,16,23,25,28);(5,8,9,7)(6,10,11,12)(13,16,17,14)(15,23,22,19)(18,25,26,20)(21,24,28,27);(6,11)(10,12)(14,16)(15,22)(20,25)(24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250360, #72256878, #72257050, …
maximal normal subgroup of	#72250360, #72256878, #72257050, …
maximal quotient of	#72264642

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6390363);