#6404646

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1856 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^c=ei,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6311842);