#454380

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#466	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2264 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)(9,10)(11,12)(13,14,18,22)(15,23,26,27)(16,21,17,20)(19,25,28,24);(5,6)(7,8)(13,15,17,25,18,26,16,24)(14,19,21,27,22,28,20,23);(2,4)(6,8)(15,24)(16,17)(19,23)(20,21)(25,26)(27,28);(10,12)(13,15,17,25,18,26,16,24)(14,19,21,27,22,28,20,23)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315807, #72315832, #72315966
maximal normal subgroup of	#72315807, #72315832, #72315966

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361576);