#499805

order	512
nilpotency class	4
isoclinism	isoclinic class of #499803
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#3027	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1995 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303827, #72312319, #72312347, …
maximal normal subgroup of	#72303827, #72312319, #72312347, …
maximal quotient of	#72303864, #72304019

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407001);