#153709

order	512
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₄⁴ · Schur multiplier (H₂) = C₂×C₄⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
commutator	C₂	C₄⁴
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

4870 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=i,b^a=be,d^a=de,c^b=ce

Conjugacy classes & character table

272 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72227444, #72228149, C₂×#72222569, …
Fitting subgroup of	#72227444, #72227445, #72227446, …
maximal subgroup of	#72211029, #72222564, #72222569, …
maximal normal subgroup of	#72211029, #72222564, #72222569, …
maximal quotient of	H(5,ℤ/4), C₂×#153709

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60905);