#6384648

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
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

1875 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)(6,9)(10,11)(13,14,19,23)(15,20,18,27)(16,25,17,22)(21,26,28,24);(5,6,8,9)(7,11,12,10)(13,15,19,18)(14,20,23,27)(16,26,17,24)(21,25,28,22);(13,16,19,17)(14,21,23,28)(15,24,18,26)(20,22,27,25);(5,7,8,12)(6,10,9,11)(13,17,19,16)(14,22,23,25)(15,24,18,26)(20,21,27,28);(1,3,4,2)(16,17)(21,28)(22,25)(24,26)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6291844);