#7198928

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996335
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 ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1951 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=fh,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=dhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257067, #72257111, #72257181
maximal normal subgroup of	#72257067, #72257111, #72257181
maximal quotient of	#72265558

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7106124);