C₂²×#2545 #7628067

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

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

12642 conjugacy classes of subgroups.

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

maximal subgroup of	C₂²×#36218, C₂²×#38078, C₂²×#44892, …
maximal normal subgroup of	C₂²×#36218, C₂²×#38078, C₂²×#44892, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7535263);