#154050

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁴
commutator	C₂³	C₂²×C₄²
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

4398 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222727, #72222729
maximal normal subgroup of	#72222727, #72222729

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 61246);