#365448

order	512
nilpotency class	3
isoclinism	isoclinic class of #24472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1463 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272644);