#221492

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	#2408	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1213 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6)(7,10)(8,11)(9,13)(12,16)(14,17)(15,19)(18,20)(21,22,24,27,25,28,31,34)(23,26,30,33,29,32,35,36);(1,2)(3,4)(5,7,9,15)(6,10,13,19)(8,14,16,20)(11,17,12,18)(21,23)(22,26)(24,30)(25,29)(27,33)(28,32)(31,35)(34,36);(6,11)(8,16)(10,17)(12,13)(14,20)(18,19)(23,29)(26,32)(30,35)(33,36);(2,4)

Polycyclic presentation:

2^9|a^2=g,b^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=de,f^c=fh

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, 128688);