#153908

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

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

5630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,b^a=be,c^a=cf,c^b=cg,d^c=de

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329950, #72330003, #72330040, …
maximal normal subgroup of	#72329950, #72330003, #72330040, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 61104);