#181417

order	512
nilpotency class	3
isoclinism	isoclinic class of #8101
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,f^2=h,g^2=i,b^a=be,c^a=ch,e^a=eh,g^a=gi,e^b=ei,f^b=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235047, #72235048, #72235049
maximal normal subgroup of	#72235047, #72235048, #72235049

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88613);