#209302

order	512
nilpotency class	3
isoclinism	isoclinic class of #23924
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	#1999	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

940 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)(5,6,11,15)(7,16,8,13)(9,14,10,19)(12,18,20,17)(21,22,25,29,26,24,28,33)(23,27,31,34,30,32,35,36);(5,7,11,8)(6,12,15,20)(9,17,10,18)(13,19,16,14)(21,23)(22,27)(24,32)(25,31)(26,30)(28,35)(29,34)(33,36);(2,4)(5,8,11,7)(6,13,15,16)(9,17,10,18)(12,19,20,14)(23,30)(27,32)(31,35)(34,36);(5,9,11,10)(6,14,15,19)(7,17,8,18)(12,13,20,16)(21,24,25,33,26,22,28,29)(23,27,31,34,30,32,35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116498);