#333117

order	512
nilpotency class	3
isoclinism	isoclinic class of #333055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

637 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 637 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ehi,c^2=hi,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=dei,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240313);