#110572

order	512
nilpotency class	3
isoclinism	isoclinic class of #9383
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

944 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 170 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239384
maximal normal subgroup of	#72239384
maximal quotient of	#72239387

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17768);