C₂²×#2275 #7625209

order	512
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄² · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂²×#2275

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator	C₂	C₂⁴×C₄²
Frattini, cosocle	C₂³	C₂⁶
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

18714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 20 points:

(1,2);(3,4);(5,6,8,7)(9,10,12,11)(13,14,17,15)(16,19,20,18);(9,10,12,11)(14,15)(18,19);(5,7,8,6)(9,11,12,10)(13,15,17,14)(16,19,20,18);(13,16)(14,18)(15,19)(17,20)

Conjugacy classes & character table

272 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×#34290, C₂²×#35921, C₂²×#34288, …
maximal normal subgroup of	C₂²×#34290, C₂²×#35921, C₂²×#34288, …
maximal quotient of	#72261552, C₂²×#13813, C₂²×D₄×#95, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532405);

C22×#2275 #7625209