#142331

order	512
nilpotency class	4
isoclinism	isoclinic class of #142216
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

731 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 731 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,c^b=cf,d^b=dg,h^b=hi,d^c=dh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240326, #72240610, #72240612
maximal normal subgroup of	#72240326, #72240610, #72240612
maximal quotient of	#72261239

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49527);