#6380666

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2106 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)(5,6)(7,10)(8,12)(9,11)(13,14,18,20,19,22,17,16)(15,21,26,27,24,28,25,23);(5,7,9,8)(6,10,11,12)(14,20)(16,22)(17,18)(21,27)(23,28)(25,26);(5,8,9,7)(6,10,11,12)(13,15)(14,21)(16,23)(17,25)(18,26)(19,24)(20,27)(22,28);(13,16,18,14,19,20,17,22)(15,23,26,21,24,27,25,28);(15,24)(21,28)(23,27)(25,26)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,d^b=df,f^b=fi,h^b=hi,e^c=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, 6287862);