#188683

order	512
nilpotency class	3
isoclinism	isoclinic class of #161262
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,f^a=fh,d^b=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95879);