#221788

order	512
nilpotency class	3
isoclinism	isoclinic class of #165149
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1431 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,f^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ch,d^b=de,f^c=fi,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 128984);