#7145778

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2451 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)(7,11)(10,12)(13,14,20,25,19,17,18,23)(15,24,26,21,27,16,22,28);(6,9)(7,11)(13,15,19,27)(14,21,17,28)(16,25,24,23)(18,26,20,22);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,22)(15,25)(17,26)(18,21)(19,24)(20,28)(23,27);(6,9)(10,12)(13,17,18,25,19,14,20,23)(15,24,22,28,27,16,26,21);(5,7,8,11)(6,10,9,12)(13,18,19,20)(14,23,17,25)(15,26,27,22)(16,28,24,21)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283906, #72284255, #72284643
maximal normal subgroup of	#72283906, #72284255, #72284643

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7052974);