#105558

order	512
nilpotency class	3
isoclinism	isoclinic class of #94972
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁶	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

2214 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 2214 conjugacy classes of subgroups — too many to draw. Only its 222 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,c^b=ch,f^b=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267923
maximal normal subgroup of	#72267923
maximal quotient of	#72268049

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12754);