#480602

order	512
nilpotency class	4
isoclinism	isoclinic class of #11549
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
socle	C₂²	C₂²×#99
prime radical	#2431	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

925 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=h,e^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ch,e^b=ei,f^b=fgi,f^e=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387798);