#7182094

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	C₂×#3257	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1806 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)(13,14,19,22)(15,21,16,17)(18,26,28,23)(20,25,27,24);(5,6,9,10)(7,8,12,11)(13,15)(14,20)(16,19)(17,23)(18,25)(21,26)(22,27)(24,28);(2,4)(5,7)(6,11)(8,10)(9,12)(13,16,19,15)(14,21,22,17)(18,25,28,24)(20,23,27,26);(6,10)(7,12)(13,17)(14,15)(16,22)(18,27)(19,21)(20,28)(23,25)(24,26);(5,8,9,11)(6,7,10,12)(14,22)(15,16)(23,26)(24,25)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,e^2=hi,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, 7089290);