#406273

order	512
nilpotency class	3
isoclinism	isoclinic class of #394267
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2411 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=f,d^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,e^c=ei,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261033, #72262061, #72262091
maximal normal subgroup of	#72261033, #72262061, #72262091

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313469);