#113110

order	512
nilpotency class	3
isoclinism	isoclinic class of #113110
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#60799	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1069 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,d^c=dhi,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214520, #72214761, #72214788
maximal normal subgroup of	#72214520, #72214761, #72214788

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20306);