#6668924

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2342 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)(15,21)(16,17)(18,20)(19,24)(22,27)(23,28)(25,26);(6,9)(7,11)(13,15)(14,19)(16,21)(17,24)(18,27)(20,28)(22,25)(23,26);(2,4)(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,22,24,28)(18,20,26,25)(19,27,21,23);(6,9)(10,12)(14,16)(19,21)(20,25)(22,28);(14,16)(15,23)(18,26)(19,28)(21,22)(24,27)

Polycyclic presentation:

2^9|c^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307449, #72307611, #72307800
maximal normal subgroup of	#72307449, #72307611, #72307800

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6576120);