#417760

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×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×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2962 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)(13,14,18,16)(15,21,23,28)(17,25,26,20)(19,27,24,22);(5,6)(7,8)(13,15,17,22,18,23,26,27)(14,19,20,21,16,24,25,28);(2,4)(6,8);(10,12)(13,16,18,14)(15,21,23,28)(17,20,26,25)(19,27,24,22)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315669, #72315717, #72316213
maximal normal subgroup of	#72315669, #72315717, #72316213

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324956);