#192014

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
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	C₂×C₄×D₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

974 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=di,e^b=eh,f^b=fh,d^c=di,e^c=eh,f^c=fh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99210);