#6470371

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470368
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 ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3094 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)(14,19)(16,17)(20,27)(21,26)(22,28)(24,25);(5,7)(6,9)(8,11)(10,12)(13,14,16,26,18,21,17,19)(15,22,24,27,23,20,25,28);(2,4)(7,11)(9,12)(13,15)(14,20)(16,24)(17,25)(18,23)(19,27)(21,22)(26,28);(7,11)(9,12)(14,21)(15,23)(19,26)(24,25);(6,10)(9,12)(13,16,18,17)(14,19,21,26)(15,24,23,25)(20,27,22,28)

Polycyclic presentation:

2^9|b^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281700, #72282222, #72282448
maximal normal subgroup of	#72281700, #72282222, #72282448

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6377567);