#522563

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3058 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)(5,6,9,7)(8,11,12,10)(14,15)(18,19)(21,22,25,23)(24,27,28,26);(1,3,4,2)(6,7)(10,11)(22,23)(26,27);(13,14,17,15)(16,18,20,19)(21,22,25,23)(24,26,28,27);(5,7,9,6)(8,11,12,10)(13,15,17,14)(16,19,20,18)(21,23,25,22)(24,27,28,26);(5,8,9,12)(6,10,7,11)(13,16,17,20)(14,18,15,19)(21,24)(22,26)(23,27)(25,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gh,d^2=fg,e^2=fgh,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	#72263755, #72263783, #72263793
maximal normal subgroup of	#72263755, #72263783, #72263793

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 429759);