#343280

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

719 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 719 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,11,15,10,7,9,13)(8,12,18,20,17,14,16,19)(21,22,25,29,26,30,35,36)(23,31,32,28,24,33,34,27);(1,2)(3,4)(5,7,10,6)(8,14,17,12)(9,13,11,15)(16,19,18,20);(2,4)(5,8,10,17)(6,12,7,14)(9,16,11,18)(13,19,15,20)(21,23,26,24)(22,27,30,28)(25,34,35,32)(29,33,36,31);(5,9)(6,7)(8,16)(10,11)(12,14)(17,18)(21,24,26,23)(22,28,30,27)(25,32,35,34)(29,31,36,33)

Polycyclic presentation:

2^9|a^2=efh,b^2=h,c^2=hi,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,e^b=eh,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250476);