#7064988

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
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
prime radical	#24444	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1946 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)(13,14,18,21,17,20,16,22)(15,23,27,19,24,25,26,28);(5,6)(7,11)(8,10)(9,12)(13,15,17,24)(14,19,20,28)(16,27,18,26)(21,23,22,25);(2,4)(5,7,8,12)(6,9,10,11)(14,20)(15,24)(21,22)(26,27);(7,12)(9,11)(13,14,16,22,17,20,18,21)(15,25,26,19,24,23,27,28);(6,10)(9,11)(13,16,17,18)(14,21,20,22)(15,26,24,27)(19,25,28,23)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=hi,d^2=f,e^2=i,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^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257058, #72257105, #72257217
maximal normal subgroup of	#72257058, #72257105, #72257217
maximal quotient of	#72265160

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6972184);