#353307

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4410 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215768, #72215990, #72216022
maximal normal subgroup of	#72215768, #72215990, #72216022
maximal quotient of	#72215720

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260503);