#510049

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

567 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=e,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343483, #72343823, #72343845
maximal normal subgroup of	#72343483, #72343823, #72343845

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417245);