#127534

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
socle	C₂³	C₂×#99
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

758 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 162 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)(5,6)(7,11)(8,10)(9,12)(14,18)(19,24);(6,10,12,11)(13,14,16,20,17,18,23,25)(15,19,21,26,22,24,27,28);(5,7,9,8)(6,11,12,10)(13,15,17,22)(14,19,18,24)(16,21,23,27)(20,26,25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266618
maximal normal subgroup of	#72266618
maximal quotient of	#72266600

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34730);