#209288

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
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	#23567	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1034 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)(7,11)(8,10)(9,13)(12,16)(14,18)(15,17)(19,20)(21,22,25,29,26,24,28,33)(23,27,32,35,30,31,34,36);(6,10)(8,16)(11,17)(12,13)(15,20)(18,19)(21,23)(22,27)(24,31)(25,32)(26,30)(28,34)(29,35)(33,36);(2,4)(5,7)(6,11)(8,15)(9,14)(10,17)(12,19)(13,18)(16,20)(23,30)(27,31)(32,34)(35,36);(7,14)(11,18)(15,20)(17,19)(21,24,25,33,26,22,28,29)(23,31,32,36,30,27,34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320625, #72320638, #72320715
maximal normal subgroup of	#72320625, #72320638, #72320715
maximal quotient of	#72321144, #72321438

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116484);