#181923

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1384 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)(13,14,16,20,17,19,24,27)(15,21,22,28,23,18,25,26);(2,4)(5,6,7,10,8,9,11,12);(13,15,16,22,17,23,24,25)(14,18,20,26,19,21,27,28);(6,9)(10,12)(14,19)(18,21)(20,27)(26,28)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=dh,d^b=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286502, #72286522
maximal normal subgroup of	#72286502, #72286522
maximal quotient of	#72286827

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89119);