#228675

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

614 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(2,5)(6,8)(9,10,14,17)(11,18,20,23)(12,16,13,21)(15,22,24,19)(25,26,29,34,30,28,31,33)(27,35,38,40,37,32,36,39);(1,2)(3,6)(4,5)(7,8)(25,27,31,36,30,37,29,38)(26,32,33,40,28,35,34,39);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21)(25,28,30,26)(27,32,37,35)(29,33,31,34)(36,40,38,39);(2,5)(6,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)(25,29,30,31)(26,33,28,34)(27,36,37,38)(32,39,35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=e,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ci,d^b=de,d^c=di,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343376, #72343868, #72343892
maximal normal subgroup of	#72343376, #72343868, #72343892

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 135871);