#522334

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=h,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263726, #72263785, #72263793
maximal normal subgroup of	#72263726, #72263785, #72263793

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 429530);