#494358

order	512
nilpotency class	4
isoclinism	isoclinic class of #33557
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator	C₂×D₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1130 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)(7,12)(8,13)(9,11)(10,14)(15,26)(16,23)(17,27)(18,30)(19,29)(20,24)(21,34)(22,25)(28,32)(31,35)(33,36);(1,3,4,2)(6,11)(8,18)(12,23)(13,28)(14,25)(15,19)(20,32)(24,30)(26,36)(27,34)(29,35)(31,33);(5,7,10,17)(6,12,14,27)(8,19,18,15)(9,21,22,16)(11,23,25,34)(13,29,30,26)(20,31,32,33)(24,36,28,35);(5,8,10,18)(6,13,14,30)(7,15,17,19)(9,20,22,32)(11,24,25,28)(12,26,27,29)(16,31,21,33)(23,35,34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312649, #72312664, #72312670, …
maximal normal subgroup of	#72312649, #72312664, #72312670, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401554);