#7182111

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	#25419	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1857 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)(6,10)(8,11)(13,14,19,23)(15,16,21,26)(17,22,18,20)(24,28,27,25);(5,6,9,10)(7,8,12,11)(14,20)(15,21)(16,25)(17,18)(22,23)(26,28);(2,4)(5,7)(6,11)(8,10)(9,12)(13,15,19,21)(14,16,23,26)(17,27,18,24)(20,28,22,25);(6,10)(7,12)(13,16,19,26)(14,21,23,15)(17,25,18,28)(20,27,22,24);(5,8,9,11)(6,7,10,12)(13,17,19,18)(14,20,23,22)(15,24,21,27)(16,25,26,28)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^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, 7089307);