#188517

order	512
nilpotency class	3
isoclinism	isoclinic class of #163899
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95713);