#333042

order	512
nilpotency class	3
isoclinism	isoclinic class of #332983
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

648 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=h,c^2=i,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=de,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240238);