#493953

order	512
nilpotency class	3
isoclinism	isoclinic class of #392203
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2280
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#20715
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1051 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214259, #72214675, #72214770
maximal normal subgroup of	#72214259, #72214675, #72214770

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401149);