#115414

order	512
nilpotency class	3
isoclinism	isoclinic class of #10038
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,e^b=eh,f^b=fh,g^b=gh,d^c=di,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22610);