#185201

order	512
nilpotency class	3
isoclinism	isoclinic class of #160892
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3511	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2373 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)(10,19)(11,15)(12,17)(18,20)(21,22,25,29,26,30,24,28)(23,31,32,27,34,35,33,36);(5,7,11,8)(6,12,15,20)(9,17,10,18)(13,19,16,14)(21,23)(22,27)(24,32)(25,33)(26,34)(28,35)(29,31)(30,36);(2,4)(5,8,11,7)(6,13,15,16)(9,17,10,18)(12,19,20,14);(5,9,11,10)(6,14,15,19)(7,17,8,18)(12,13,20,16)(21,24,26,25)(22,28,30,29)(23,32,34,33)(27,35,36,31)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92397);