#181900

order	512
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂⁴
commutator, socle	C₂³	C₂²×C₄²
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1666 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286025, #72318433, #72319546, …
maximal normal subgroup of	#72286025, #72318433, #72319546, …
maximal quotient of	#72286127, #72286354, C₂×#181900

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89096);