C₂²×#2677 #7633450

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

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

12034 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)(13,14,17,19)(15,16,20,18);(10,12)(13,15,17,20)(14,18,19,16);(5,6)(7,8)(9,10)(11,12);(5,6)(7,8)(13,16,17,18)(14,15,19,20)

maximal subgroup of	C₂²×#52096, C₂²×#43543, C₂²×#59122, …
maximal normal subgroup of	C₂²×#52096, C₂²×#43543, C₂²×#59122, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540646);