#332990

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	#3496	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 752 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) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2,4,7)(3,5,8,6)(9,10,15,20,16,11,14,19)(12,21,24,13,18,22,23,17);(9,11,16,10)(12,22,18,21)(13,23,17,24)(14,19,15,20);(2,5)(3,8)(6,7)(9,12)(10,17)(11,13)(14,23)(15,24)(16,18)(19,22)(20,21);(2,5)(3,8)(6,7)(9,13)(10,18)(11,12)(14,22)(15,21)(16,17)(19,23)(20,24)

Polycyclic presentation:

2^9|a^2=ei,b^2=h,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, 240186);