#494056

order	512
nilpotency class	4
isoclinism	isoclinic class of #33688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3109
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#20738
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

934 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286402, #72286539, #72286542
maximal normal subgroup of	#72286402, #72286539, #72286542

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401252);