#7148401

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#30859	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1803 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7055597);