#7198980

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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1975 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:

(2,4)(5,6)(7,10)(8,11)(9,12)(13,14,17,22,18,23,16,21)(15,24,25,28,26,20,27,19);(5,6)(7,10)(8,11)(9,12)(13,15)(14,19)(16,25)(17,27)(18,26)(20,22)(21,24)(23,28);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,15)(14,20)(16,27)(17,25)(18,26)(19,22)(21,28)(23,24);(5,8,10,9)(6,11,7,12)(13,14,16,21,18,23,17,22)(15,20,27,28,26,24,25,19);(5,9,10,8)(6,11,7,12)(13,16,18,17)(14,21,23,22)(15,25,26,27)(19,24,28,20)

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=fhi,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	#72257105, #72257134, #72257220
maximal normal subgroup of	#72257105, #72257134, #72257220
maximal quotient of	#72308480

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7106176);