#182013

order	512
nilpotency class	3
isoclinism	isoclinic class of #17392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1568 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=i,g^2=i,b^a=be,c^a=ci,g^a=gh,d^b=di,e^b=eh

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, 89209);